Equilibria and precession in a uniaxial antiferromagnet driven by the spin Hall effect

We systematically study the stationary and precessional states in a uniaxial antiferromagnet under the dampinglike spin–orbit torque (SOT). The stable regions for all equilibria are defined by the linear stability analysis. In the regions without any stable equilibrium, we confirm that a stable conical precession may exist. Invoking symmetry arguments, we rigorously reduce the coupled Landau–Lifshitz–Gilbert equations to a single-vector one, which allows us to derive the analytic expressions of the lower and upper thresholds, the frequency, and the amplitude of precession for a weak uniaxial anisotropy. Its frequency is of the order of THz and increases almost linearly with the current. Further analysis reveals that the precession is mainly propelled by the exchange interaction in the promise that the SOT balances the damping one in average. Moreover, the investigation uncovers that a weak anisotropy can improve the frequency tunability, and a small damping benefits lowering the exciting current. Finally, the analytic expressions are verified by comparing with numerical simulations of the original Landau–Lifshitz–Gilbert equations.

In order to elucidate the key physical features of the AFM oscillator, such as the threshold, frequency, and amplitude, an analytic solution is indispensable. However, the AFM Landau-Lifshitz-Gilbert (LLG) equations are too complicated to be dealt with analytically. A usual approximate analytic method is the l-m scheme [6-9, 11-15, 17], In which, the coupled LLG equations of the sublattice magnetization are firstly transformed into another coupled equations about the Néel vector l and the average magnetization m. Then, taking the strong exchange limit and the approximation |m| |l|, the coupled l-m equations are reduced to a single-vector equation of l with m being a slave vector. Because the reduced l-equation is decoupled with m, it is easily treated analytically. According to this scheme, the AFM dynamics under spin torques have been reduced to a nonlinear forced vibration [6,8,14], which has been used to investigate the precession and switching more succinctly. In these cases, the sublattice moments remain nearly antiparallel, so that the approximation |m| |l| is satisfied. However, for general cases, such as the conical precession, m may be sizable comparing with l. For example, the numeric results of this paper (figures 2-4) reveal that,

Model
We consider an AFM thin film attached with a heavymetal (HM) layer, as shown in figure 1. With a longitudinal electric current flowing through the HM layer, the dampinglike SOT is generated via the spin Hall effect [16]. Under the SOT, the magnetic dynamics in AFM layer is governed by two coupled sublattice LLG equations, where m i is the unit vectors of magnetization in two sublattices marked by i = 1, 2. α is the Gilbert constant of damping. Here, the inhomogeneous exchange contribution is ignored. So, we focus on the magnetic dynamics within the framework of a two-macrospin model [15], which is a reasonably good approximation for the small-size sample. Then, including the exchange and uniaxial anisotropy terms, the magnetic energy reads, E = ω ex m 1 · m 2 − 1 2 ω an (m 1 · e a ) 2 + (m 2 · e a ) 2 , with e a being the unit vector along the easy axis. Here, the demagnetization energy is not considered. This can be justified by the fact that the demagnetization along the film normal is dominant. In the following sections, we find three states under the SOT, including the tilted-AFM, the precession, and the e p -FM, for  which the components of m along the film normal are all zero. So, the demagnetization effect (shape anisotropy) is negligible.
The dampinglike SOT is expressed as with e p being the unit vector along the spin polarization. All parameters related with the strengths of torques have been scaled with frequency. ω ex = γH ex with γ being the gyromagnetic ratio and H ex the inter-sublattice exchange field. ω an = γH an with H an being the anisotropy field. ω sot = u/d with d being the thickness of AFM layer. u has the dimension of velocity, u = μ B /(eM s )ξj e with μ B being the Bohr magneton, e the element charge, M s the sublattice saturation magnetization, and j e the electric current density. ξ is the SOT efficiency which equals to T int θ sh [37,38], with θ sh being the spin Hall angle, and T int the spin transparency of the interface [39]. Without the SOT, from the static equation m i × dE/dm i = 0, it is easy to get the equilibria. Their stabilities can be judged by the linear stability analysis. Table 1 lists all the equilibria of m 1,2 and their stability types. When applying the SOT, do these equilibria remain their stability? Do new equilibria and dynamic states emerge? In the following, we will investigate these issues for two configurations: the easy axis perpendicular and parallel to the spin polarization.

Linear stability analysis
Because the AFM LLG equations are coupled ones about four independent variables, the linear stability analysis is somewhat different from the ferromagnetic counterpart. In this section, we present a general method to judge the stability of equilibria for bipartite AFM systems. Due to |m i | = 1, it is convenient to parameterize m i in terms of the polar angle θ i and the azimuthal one φ i according to m i = (sin θ i cos φ i , sin θ i sin φ i , cos θ i ). Then, the coupled LLG equations (equation (1)) can be written in the form of a 4 × 4 matrix, where x = (θ 1 , φ 1 , θ 2 , φ 2 ) T , with T denoting the matrix transpose, and ∂ x = (∂/∂θ 1 , ∂/∂φ 1 , ∂/∂θ 2 , ∂/∂φ 2 ) T . In addition, A = diag(A 1 , A 2 ), with The SOT term is expressed as , where the spherical unit vectors e i θ = (cos θ i cos φ i , cos θ i sin φ i , − sin θ i ), and e i φ = (− sin φ i , cos φ i , 0). The equilibrium directions of m 1,2 can be obtained by setting dx/dt = 0. The solutions are denoted by T . Applying small perturbations, the system deviates from the equilibrium state slightly.
Namely, x = x 0 + x , where x = θ 1 , φ 1 , θ 2 , φ 2 T is regarded as a response to the perturbations. Inserting this ansatz into equation (4), and linearizing it in the vicinity of equilibria, one can get where where and In addition, the matrix T is related with the SOT, which is written as T = diag(T 1 , T 2 ), where It should be mentioned that this 4 × 4 matrix description of the linearized AFM dynamics has been used to rewrite the spin wave Hamiltonian of AFM [40]. This treatment avoid taking the approximation |m| |l|. Our formalism includes the nonconservative terms, such as the SOT and the damping. Usually, the solutions of equation (6) take the form x ∝ e λt . To ensure the existence of nontrivial solutions, λ satisfies the secular equation, Generally, this secular equation can be expanded as a quartic one, where the parameters are determined by the equilibrium solutions (θ 0 i , φ 0 i ). If all the roots of λ have a negative real part, the corresponding equilibrium state is stable. According to Routh-Hurwitz criterion [18][19][20][21], one can define a series of determinants, If all Δ are positive, the real parts of all roots of λ are negative. Namely, one gets a stable equilibrium state. In the following sections, we will use this formalism to seek the stable equilibria of AFM system driven by the dampinglike SOT.

Case that e p ⊥e a
In this section, we obtain all the equilibria for the case with the spin polarization (e p ) perpendicular to the easy axis (e a ) and analyze their stability. By Melnikov's method, the thresholds, the frequency, and the amplitude of stable precession are also calculated analytically.

Equilibria and their stabilities
The equilibria are defined if the SOT balances out the precession torques. So, setting dx/dt = 0 in equation (4), and taking e a = e y and e p = e z (as shown in figure 1), the equilibrium equations can be obtained, where i = 1, 2, denoting the two sublattices. Firstly, it is easy to observe that sin θ 0 i = 0 solve equations (17) and (18). This generates two FM states that θ 0 1 = θ 0 2 = 0 or π, and two AFM ones that θ 0 1 = 0 (π) and θ 0 2 = π (0). For these two kinds of equilibria, because m 1 Secondly, for sin θ 0 i = 0, from equation (17) one has Equation (21) allows φ 0 1 = φ 0 2 or φ 0 1 = ±π + φ 0 2 . Inserting these two relations into equation (18), one has the solution θ 0 1 = θ 0 2 = π/2. From equation (20), we can get φ 0 with P = 1, 2, 3, 4. These two kinds of equilibria indicate that m 1,2 remain parallel (FM) or antiparallel (AFM) in the x-y plane. Based on this characteristic, we name them after tilted FM and tilted AFM states. Stability analysis reveals that the tilted FM states are unstable (see appendix A4 for the detailed derivations). The tilted AFM states are unstable for P = 1, 3, while stable for P = 2, 4 under the condition |ω sot | < ω an /2 (see appendix A3 for the detailed derivations). The formation of the stable tilted AFM state can be illustrated by a simple argument. The SOT pushes m 1,2 deviate from the easy axis (y axis), then the exchange and anisotropy torques rotate m 1,2 around the easy axis. When m 1 and m 2 rotate to the x-y plane, they restore antiparallel. In this scene, the exchange torques vanish. In a proper current region, the SOT is balanced by the anisotropy torque, and the tilted AFM state emerges. For clarity, we list all the equilibria and their stabilities in table 2. Comparing with the equilibria listed in table 1, one can find that, under the SOT with e p ⊥e a , the AFM and FM states along the easy axis do not exist anymore, to be replaced by the novel tilted AFM and FM states. The AFM and FM states normal to the easy axis are also not equilibria except that m i lie along the spin-polarization direction.
For the uniaxial AFM with a weak anisotropy, such as MnF 2 , ω u sot > ω an /2. Therefore, by linearization method, there is no equilibrium in the regions that ω an /2 < |ω sot | < ω u sot . By integrating the coupled LLG equations (equation (1)), it can be found that the precession may emerge in this region (see figure 3). In the next subsection, the precession state is analyzed by combining the analytic and numeric methods.

Reduced equation
In general, the two coupled LLG equations are too complex to be solved analytically. Then, in many works [6-9, 11-15, 17], equation (1) is transformed into the coupled equations in terms of the uncompensated Table 2. Equilibria and their stable regions in the presence of SOT. Here, Ψ = arcsin(2ω sot /ω an ), and ω u sot and φ 0 are expressed by equations (19) and (22) [41,42]. Here, the inter-sublattice exchange field H ex = 526 kOe, and the anisotropy field H an = 8.2 kOe. Then, ω ex = 9.25 × 10 12 rad s −1 and ω an = 1.44 × 10 11 rad s −1 . Here, we adopt the typical value of damping constant α = 0.01 as an illustrating example. To get this tilted AFM state, we take ω sot = 0.391ω an . magnetization m = 1/2(m 1 + m 2 ) and the Néel order parameter l = 1/2(m 1 − m 2 ). Under the assumption that |m| 1 and |l| ≈ 1, the motion of l is decoupled from that of m. So, the transformed equations are simplified as a nonlinear dynamic equation about l. m becomes a slave variable and can be derived from l. The validity of this assumption is seldom concerned [15]. In fact, when m 1,2 deviate from the antiparallel state greatly, this assumption may not be satisfied.
Before dealing with equation (1) analytically, we solve them numerically for different strengths of SOT by the solver ode45 of Matlab. Setting the initial values that m 1,2 are antiparallel and along the easy axis, we get the arrays of components of m 1,2 at a time sequence for different strengths of SOT. Figures 2-4 (and figure 7 in next section) are plotted by this data directly. In these figures, we summarize the evolutions of components of m 1,2 (the magnetization of two sublattices), m (the uncompensated magnetization), and l (the Néel vector), as well as the magnitudes of m and l. Without the SOT, the AFM state is stable with m 1,2 along the easy axis, as listed in table 1. Applying the SOT, they are deviated from the easy axis, but remain in the x-y plane and antiparallel for the weak SOT, as shown in figure 2. Increasing the strength of SOT, m 1,2 are tilted out of the x-y plane and precess out of phase around the spin polarization (z axis), as shown in figure 3. Increasing the strength of SOT further, m 1,2 are oriented along the spin polarization and the system enters e p -FM state, as shown in figure 4.
From these numeric results, several important conclusions can be obtained. Firstly, comparing figures 2 and 3, one can find that the threshold of ω sot is about 0.392ω an for the switching from the tilted AFM state to the precession. This is less than 1/2ω an , the critical value to destabilize the tilted AFM state. Secondly, observing figures 3(f) and 4(f), the conditions |m| 1 and |l| ≈ 1 are unsatisfied for the precession and the e p -FM state. The deviations of |m| from 0 and |l| from 1 increase with ω sot increasing. So, it is impossible to decouple the Néel vector l and the uncompensated magnetization m. The usual l-m scheme  are reduced as where the reduced magnetic energy and the SOT So, a bipartite AFM with a uniaxial anisotropy is translated into a single-order-vector magnetic system with a biaxial anisotropy. Apart from the easy axis along e y , there exists an easy plane normal to e p and determined by the AFM exchange interaction. Because this reduced dynamic system is two-dimensional, it can be determined that there exist non-decaying precessional states in the region without stable equilibria. This is a logical consequence of Poincaré-Bendixson theorem [22,43]. From figures 2 and 3, we have find that the lower threshold of precession is ω l sot ≈ 0.392ω an , which is less than the critical value (1/2ω an ) to destabilize the tilted AFM state. Thus, the linear analysis cannot predict the details of this precessional state. In the following, based on equations (23)-(25), we will use the Melnikov's method to redefine the lower and upper thresholds of precession, and to calculate its frequency and amplitude.
Melnikov's method is suited to the weakly perturbed conservative systems [22] and has been successfully applied on the ferromagnetic systems driven by spin torques [24]. Here, due to the strong exchange interaction, the damping and the SOT can be treated as small perturbations. When the energy influx from the SOT balances the damping dissipation, a stable precession can be excited, which is approximately along some constant-energy trajectories derived from the energy landscape defined by equation (24).

Constant-energy trajectories
Considering the conservation of modulus, the vector n evolves on the surface of the unit sphere |n| = 1. So, the unperturbed trajectories can be defined as the intersection of the elliptic cylinders determined by equation (24) with this unit sphere. These trajectories are identified by E n . In addition, there are three equilibria defined by equation (24). The first one is that n is oriented along the easy axis (y-axis). The corresponding energy is minimal and defined as E min = −1/2(ω ex + ω an ). The second one is a saddle point, for which n is perpendicular to the easy axis and the spin polarization, as marked in figure 5(b). The corresponding energy E saddle = −1/2ω ex . The third one is that n points along the spin polarization (z-axis). The corresponding energy is maximal and defined as E max = 1/2ω ex .
The trajectories through the saddle points act as separatrixes, which are defined as the locus of intersection of two planes (determined by n y = ± 2ω ex /ω an n z ) with the unit sphere, as shown by the red solid curves in figure 5(a). For a weak anisotropy (ω an ω ex ), these trajectories are very close to the x-y plane.
When E min < E n < E saddle , n rotates around the easy axis (y-axis), as represented by the green dotted curves in figure 5(a). These trajectories are named after low-energy ones. Here, the spin polarization e p points outside of the trajectory. With that in mind, projecting the SOT along the direction of damping torque, it can be find that the component of SOT is parallel to the damping torque in a half trajectory, whereas antiparallel in another half [44,45]. Therefore, it is impossible to balance the energy gain from the SOT and the energy dissipation from the damping during a whole precession. There cannot exist stable precessions on the low-energy trajectories.
When E saddle < E n < E max , n rotates around the spin polarization (z-axis), as represented by the blue dashed curves in figure 5(a). For these high-energy trajectories, the SOT may oppose the damping torque during the entire precession. Thus, a stable precession is possible. For these trajectories, it is convenient to parameterize equation (24) as where η varies from 0 to 2π, and

Balance between SOT and damping
For a high-energy trajectory (denoted by Γ), the dissipative energy by the damping torque during an entire precession is Inserting the components of equation (23) into equation (30), and by use of the parametrization of trajectory (equations (26) and (27)), the integral can be calculated analytically. Keeping only the linear terms of α and ω sot , the dissipative energy reads where E and K denote the complete elliptic integral of the second and first kinds, and the modulus In addition, around the same trajectory, the pumped energy by the SOT during a whole precession is calculated as Equating W damp and W sot , the SOT strength ω sot necessary to excite a stable precession on a constant-energy trajectory marked by E n can be expressed as, It is worth noting that, for ω sot > 0, the trajectories are on the half sphere with m 1z (m 2z ) > 0. m 1,2 rotate left-handedly around the z axis, as labeled by the blue arrow in figure 5(a). This results in that the damping torque points away from the rotational axis. The SOT is just the reverse. So, the balance between both of them may be realized during a whole precession. Moreover, the rotation is mainly propelled by the exchange interaction [6]. If ω sot < 0, the trajectories are on the hemisphere with m 1z (m 2z ) < 0 and the precessional direction is reversed. Setting E n = E max in equation (34), one can get the upper threshold for precession, which is just ω u sot (equation (19)). This is consistent with the result of linear stability analysis of e p -FM state. Taking the limit that E n → E saddle , one has another lower threshold Taking the parameters in the caption of figure 2, the value of this lower threshold is about 0.072ω an , which is much less than the numeric result 0.392ω an . This indicates that, in common with the linearization method, the Melnikov method also fails in the transitional region between the tilted AFM state and the precession.

Lower threshold of precession
To derive the lower threshold analytically, we use a method developed by Taniguchi [33], which gives an analytic result consistent with the numeric calculation very well. In figure 5(b), we plot the evolution of n at the lower threshold ω sot = 0.392ω an . Initially, n is oriented along the positive y direction (point F on the sphere) for ω sot = 0. The plot indicates that before entering the high-energy trajectory, n must evolve from the initial stable state to the saddle point (point S on the sphere). This part is not a constant-energy trajectory. To reach the saddle point, n needs to climb over an energy barrier E saddle − E min = 1/2ω an . Namely, apart from balancing the damping torque, the SOT should also do work to overcome this barrier. This work-energy relation can be expressed as, It is difficult to solve this equation analytically, because there is no explicit express for the trajectory from the stable focus (F) to the saddle point (S). However, due to ω an ω ex , the trajectory of E saddle is very close to the x-y plane. Therefore, as show in figure 5(b), the integral trajectory can be approximated by the constant-energy one of E saddle from F ± to S. F ± situate in the y-z plane with n z = ± ω an /(2ω ex + ω an ) and n y = 2ω ex /(2ω ex + ω an ).
Then, by the similar method used in obtaining equations (34) and (35), integrating equation (36) (after replacing the lower limit F by F + or F − ), a lower threshold of precession can be derived, Taking the parameters in the caption of figure 2, ω l sot ≈ 0.3918ω an , which well reproduces the value of the lower threshold of precession estimated from the numeric simulation.

Frequency
For the high-energy trajectories, the SOT balances with the damping torque in average during a precession. So, the period (frequency) is mainly determined by the unperturbed parts of equation (23), which are written in the Cartesian coordinate system as dn x dt = 2ω ex n y n z , dn z dt = ω an n x n y .
Substituting equations (26) and (27) into one of above equations, and integrating over the whole trajectory, the period can be calculated as a function of E n , The final result (frequency) is expressed by the elliptical integral, where K denotes the complete elliptic integral of the first kind, and the modulus k is still equation (32). When E n = E saddle , f = 0. When E n = E max , f = 1/(2π) √ 2ω ex (2ω ex + ω an ), which is just the resonance frequency of e p -FM states.
Removing E n from equations (34) and (42), one can get the dependence of frequency on ω sot . But, there is no explicit expression. We plot f-ω sot curve in figure 6(a). There is a good agreement between the numerical simulation (red stars) and the analytical theory (blue solid line) in the region between ω l sot and ω u sot . Below ω l sot , the numerical simulation indicates that the tilted AFM state is stable, as shown in figure 2. Obviously, the frequency increases almost linearly with ω sot . In the absence of uniaxial anisotropy (ω an = 0), f = 1/(2π)ω sot /α.  (19), (35) and (37), respectively. Here, we use ω an to scale the SOT strength ω sot because the instability occurs when the SOT overwhelms the anisotropy, and the lower threshold is comparable to the value of ω an .
It is interesting to discuss the relation between the material parameters and the frequency tunability. From equations (34) and (42), it is easy to observe that the frequency is a function of ω sot /α. So, to excite a precession with a certain frequency, small damping can decrease the required current density. Additionally, from equations (19) and (37), it can be proved that d(ω u sot − ω l sot )/dω an < 0. Namely, the current window gets narrow with ω an increasing, so does the frequency window. In the limit of weak anisotropy, from equation (37) the lowest frequency is about 1/π(1 + 2α 2ω ex /ω an )ω an /α. There is no simple relation with the material parameters. On the other hand, equation (19) indicates that the upper threshold of ω sot is proportional to α. Then, the highest frequency, which is about 2ω ex , mainly depends on the exchange interaction.

Amplitude
Removing E n from equations (28), (29) and (34), one can obtain the dependence of amplitudes of n x and n y on ω sot . In figure 6(b), the result from the analytic formula is displayed as the solid lines. The validity of the derived formula can be confirmed by comparison with a numeric integration of the coupled LLG equations (equation (1)). With ω sot increasing, the amplitudes decrease and n is directed along the spin polarization at the upper threshold. This means that the conical angle of precession decreases with ω sot increasing.

Case that e p e a
For this configuration, the system has the axial symmetry which makes the analysis easier than the case that e p ⊥e a .

Equilibria and their stabilities
Here, e a = e p = e z , as shown in figure 1. The equilibria are determined by the balance between the SOT and the precession torques produced by the exchange and anisotropy fields. Taking dx/dt = 0 in equation (4), the equilibrium equations read, where i = 1, 2, denoting the two sublattices. From equation (43), one can derive that sin 2 θ 0 1 + sin 2 θ 0 2 = 0. This means that sin θ 0 1,2 = 0 which also satisfy equation (44). So, there are four equilibria: θ 0 1 = 0 and θ 0 2 = π, θ 0 1 = π and θ 0 2 = 0, θ 0 1,2 = 0, as well as θ 0 1,2 = π. The first two correspond to degenerate AFM states. The last two are FM states. The stability can be determined by the linearization method and the Routh-Hurwitz criterion which have been formulated in section 3. The detailed derivations are presented in appendix B. The AFM states are stable under the condition that |ω sot | < ω l sot with ω l sot = α ω an (2ω ex + ω an ).
The FM state with θ 1,2 = 0 (π) is stable under the condition that ω sot > ω u sot (ω sot < −ω u sot ), where All the equilibria have been found. So, in the region that ω l sot < |ω sot | < ω u sot , there is no stable equilibrium and a precession possibly emerges.

Precession
Analogous to section 4.2.1, in view of the symmetry and the numeric results (for example, figure 7), a reduced equation like equation (23) is obtained from equation (1) by setting m 1x = −m 2x = n x , m 1y = −m 2y = n y and m 1z = m 2z = n z . These relations can also be observed in figure 7. Only the reduced magnetic energy is different from the case of e p ⊥e a , which reads, Considering ω ex > ω an generally, a bipartite AFM with e a e p is translated into a single-order-vector magnetic system with an easy plane perpendicular to the spin polarization (e p ). Due to the axial symmetry, the constant-energy trajectories are just the latitude lines on the unit sphere and there exists an analytic solution of precession. It is not necessary to resort to Melnikov's method. In order to solve equation (23) for the case of e a e p , the vector n is conveniently parametrized in spherical coordinates as n = (sin θ cos φ, sin θ sin φ, cos θ). Then, equation (23) with the magnetic energy equation (47) is transformed into The functions on the right of equations (48) and (49) are independent of φ. In order to look for the solutions, it is convenient to remove dφ/dt from equations (48) and (49). Then, one has For this θ-equation, there exist two kinds of equilibria. The first one is θ 0 = 0 (π), which is stable for ω sot > ω u sot (ω sot < −ω u sot ). These stable conditions are derived in appendix C by linear stability analysis of equation (50). These two equilibria are exactly the FM states illustrated in section 5.1.
The other equilibrium of equation (50) is which is stable for |ω sot | < ω u sot (see appendix C). For this stable equilibrium of θ, one can obtain the velocity of φ from equation (48), This is also the circular frequency of precession. Based on these solutions, the precessions of m 1,2 can be described by These analytic expressions are consistent with the numerical results shown in figure 7, supporting the validity of above argument. Unlike the case of e p ⊥e a , here the frequency window can be given analytically, which ranges from √ ω an (2ω ex + ω an ) to 2ω ex + ω an . This is easily obtained from equations (45), (46) and (52). The frequency window is mainly determined by the exchange interaction and the magnetic anisotropy, independent of the damping. But, a small damping favors decreasing the exciting current. The frequency window is shrunken if increasing ω an , because d(ω u sot − ω l sot )/dω an < 0. When ω an > 2/3ω ex , the window closes, i.e. there is no precession. Therefore, a small damping and a weak anisotropy benefit the frequency tunability.
Furthermore, the characteristics of precession are summarized in figure 8. For ω sot > 0, equation (51) means that θ 0 < π/2. Namely, the precession trajectories are above the equator. From equation (52), dφ/dt < 0. So, the precession direction obeys the left-hand thumb rule when the thumb points along the positive z-direction, as marked by the arrowhead on the upper trajectory (dotted red line) in figure 8. For this precession direction, the damping torque makes m 1 deviate from the easy axis. But, the SOT drives m 1 approaching the easy axis. The balance between these two torques enables the stable precession. On the other hand, along the tangential of this trajectory, there exist an exchange torque ω ex m 1 × m 2 and an anisotropy torque −ω an (m 1 · e a )(m 1 × e a ). As indicated by the tangential arrows at the endpoint of m 1 , the exchange torque makes m 1 rotate left-handedly around the axis. While, the anisotropy torque makes m 1 rotate right-handedly around the axis. Because the exchange is stronger than the anisotropy, m 1 rotates left-handedly. This is very different from the ferromagnet, for which the anisotropy field determines the sense of rotation and the left-hand rotation is not allowed.

Discussions
In this section, several instructive remarks are in order. Firstly, it is helpful for application to estimate the thresholds and the frequency of precession. By the definition of ω sot , the corresponding current density can be calculated from j = (eM s d)/(ξμ B )ω sot . The saturation magnetization M s ≈ 47.7 kA m −1 [42]. Considering the experimentally feasible parameter, the SOT efficiency ξ = 0.32 [46], and the thickness of AFM layer d = 4 nm. Then, by use of the magnetic parameters in the caption of figure 2 and equations (19), (37), (45) and (46), the lower and upper thresholds of current density can be calculated. For the case e p ⊥e a , the precession happens with the current varying from 5.82 × 10 7 A cm −2 to 1.92 × 10 8 A cm −2 . The corresponding frequency varies from 0.90 THz to 2.97 THz. For the case e p e a , the precession happens with the current varying from 1.69 × 10 7 A cm −2 to 1.89 × 10 8 A cm −2 . The corresponding frequency varies from 0.26 THz to 2.92 THz. These frequencies lie in the range of terahertz radiation. To decrease the exciting current, one can use the material with a smaller damping. The adjustable range for e p ⊥e a is slightly smaller than that for e p e a . This is because the SOT must overcome the energy barrier of uniaxial anisotropy before the stable precession happens for e p ⊥e a . MnF 2 is a nearly ideal uniaxial antiferromagnet. But, the low Néel temperature is unfavorable for application. So, a uniaxial antiferromagnet with higher critical temperature is anticipated.
Secondly, it should be emphasized that no approximation is taken when reducing the coupled LLG equations. This is different from the usual l-m scheme, in which, to decouple the stagger magnetization l and the average magnetization m, too many terms related with the damping and spin torques were discarded. This results in omission of some dynamic properties. Therefore, entering the nonlinear regime, one has to be a bit careful when applying the l-m scheme on a spin torque-driven antiferromagnet. Moreover, if taking ω an → 0, the two cases considered above become identical. Both the lower threshold of ω sot (equation (37)) for e p ⊥e a and the one (equation (45)) for e p e a become zero. Both the upper threshold of ω sot (equation (19)) for e p ⊥e a and the one (equation (46)) for e p e a become 2αω ex . In this limit, the frequencies are also the same, taking 1/(2π)ω sot /α. These facts also justify our analytical treatment.
Thirdly, our results indicate that a stable AFM precession is free from the magnetic field. A relatively strong dampinglike SOT tilts the sublattice magnetization m 1,2 deviating from the antiparallel state. Then, m 1,2 are exposed to the action of exchange torque, which propels them precessing. When the SOT balances the intrinsic damping torque, the precession can exist stably. Unlike the ferromagnet, it is the exchange interaction that drives the precession for AFM.
Fourthly, it should be pointed out that the fieldlike SOT is ignored. Including this fieldlike term −βω sot m 1,2 × e p with β being the relative strength of fieldlike SOT to the dampinglike one, the calculations indicate that β enters as a factor 1 + αβ before ω sot for the threshold values and the frequency-current relations. For typical experimental parameters, αβ 1. In addition, for the case of e p ⊥e a , the fieldlike SOT changes the tilted AFM states slightly by cocking m 1,2 up from the x-y plane with the tilted angle π/2 − arccos[2βω sot /(2ω ex + ω an + ω 2 an − 4ω 2 sot )]. In the large-exchange limit, this angle is vanishingly small. Beyond these two points, although the field-like SOT make the calculations more lengthy, it nearly has no influence on the final results. Moreover, for the SOTs induced by the spin Hall effect, the dominant component is dampinglike [16]. Therefore, we neglect the field-like SOT.
Finally, it is worthwhile to compare with two similar previous works [8,15]. In reference [15], a uniaxial AFM is studied under the spin torques with arbitrary spin polarizations. During the precession, two sublattice magnetic moments are assumed to remain antiparallel. Moreover, their precessions are restricted in a plane normal to the spin polarization. However, one can find that, by observing the evolutions of magnetization (for example, figure 7), this is just a special case near the lower threshold. In reference [8], several similar results, such as the lower threshold and the frequency-SOT relation, were presented for the case of e p ⊥e a . The lower threshold of reference [8] can be obtained by taking the limit ω an /ω ex → 0 in equation (37). Meanwhile, the frequency-SOT relation in reference [8] can be gotten from equations (34) and (42) by taking ω an → 0. The upper threshold of precession is not concerned in this work. In addition, this work only consider a special case that the precession was confined in the plane normal to the spin polarization by a strong easy-plane anisotropy. More universally, our work demonstrates that the precessional magnetic moments form a conical surface with an adjustable cone angle.

Conclusion
We have investigated the stationary and dynamic states of the AFM films with a uniaxial magnetic anisotropy driven by the dampinglike SOT with the spin polarization perpendicular and parallel to the easy axis. The stabilities of all equilibria are analyzed by the linearization method and Routh-Hurwitz criterion. Based on a reduced single-vector equation, we obtain the analytic expressions of the thresholds, the frequency, and the amplitude of the stable precessions. These results are verified by numerically integrating the AFM LLG equations.
For the case with the spin polarization perpendicular to the easy axis, if |ω sot | < ω l sot , the system is in a state of tilted AFM. If |ω sot | > ω u sot , the system is in a ferromagnetic state. The stable precession occurs in the region that ω l sot < |ω sot | < ω u sot . The analytic formula of the lower threshold ω l sot (equation (37)) and the upper one ω u sot (equation (19)) have been derived. For the case with the spin polarization parallel to the easy axis, the system remains AFM state provided |ω sot | < ω l sot , while enters the ferromagnetic state when |ω sot | > ω u sot . In the middle region that ω l sot < |ω sot | < ω u sot , a stable precession emerges. Also, we derive the analytic formula for the lower threshold ω l sot (equation (45)) and the upper one ω u sot (equation (46)). For both cases, it is mainly the exchange interaction to propel the precession on the promise of an average balance between the SOT and the damping. The precessional frequency is of the order of a few THz, and increases almost linearly with the strength of SOT. Moreover, a weak anisotropy favors improving the frequency tunability through enlarging the frequency window. A small damping favors reducing the current to excite a precession. Our results not only enrich the study of current-driven AFM dynamics, but also provide a solid clue for designing the AFM-oscillator with a uniaxial magnetic anisotropy.
According to the Routh-Hurwitz criterion, if a 1 , a 2 , b 1 , and b 2 are all positive, the equilibrium state is stable. For a realistic AFM, the exchange coupling is generally stronger than the anisotropy, i.e. ω ex > ω an > 0. So, it is easy to conclude that the stability condition is ω sot > ω u sot , with Similarly, the stability condition for the e p -FM state with m 1,2 contrary to the spin-polarization direction (φ 0 1 = φ 0 2 = −π/2) can also be deduced as ω sot < −ω u sot .

Appendix B. Stability analysis of the equilibria for the case that e p e a
In this appendix, using the linear stability analysis method illustrated in section 3, we derive the expressions for the stable regions of AFM and FM states presented in section 5.1. Similar to appendix A1, to avoid the illness of equilibria such as θ = 0 or π, we chose the coordinate system with x axis along the easy axis (e a ), y axis along the spin-polarization direction (e p ), and z axis normal to the film.