Spin-transfer torque induced spin waves in antiferromagnetic insulators

We explore the possibility of exciting spin waves in insulating antiferromagnetic films by injecting spin current at the surface. We analyze both magnetically compensated and uncompensated interfaces. We find that the spin current induced spin-transfer torque can excite spin waves in insulating antiferromagnetic materials and that the chirality of the excited spin wave is determined by the polarization of the injected spin current. Furthermore, the presence of magnetic surface anisotropy can greatly increase the accessibility of these excitations.


I. INTRODUCTION
As a scientific enterprise, the role of spintronics is to investigate and organize phenomena concerning spin angular momentum. 1 Of much recent interest in this field are spin-transfer torque, 2,3 spin pumping, 4 current-induced magnetization dynamics (spin waves), 5 the (inverse) spin Hall effect, 6,7 and the more recent spin caloritronics. 8 The understanding of (pseudo)spin related phenomena in graphene and topological insulators is likewise topical. 9 Yet spin angular momentum was one of the first measured quantum effects 10 and makes contact with nearly every modern subset of quantum mechanics. As a technological enterprise, this rich intersection of spintronic physics hosts a vast and non-trivial dynamical landscape with many yet-unexplored avenues of research. The degree to which spintronics can be applied to problems in information architecture is already quite promising, and it is likely that the full extent of these technologies is presently unrealized.
To date, almost all research on spintronics has focused on ferromagnetic (FM) materials with comparatively little attention given to antiferromagnets (AFM). This is because AFM order has no net magnetization, which makes it difficult to detect and control with magnetic fields. Largely for this reason, technological applications of FM materials are considerably more prevalent than those of AFM: at present, the most important and perhaps the only use of AFM materials is the use of exchange bias for pinning the FM magnetization. 11 Consequently, there exists a significant gap in knowledge concerning spintronics in AFM materials. On the other hand, the lack of coupling between magnetic fields and AFM order also makes AFM robust against undesirable external magnetic perturbations, and consequently immune to harsh conditions such as natural or man-made electromagnetic pulses. Spintronics applications in AFM could inherit this desirable trait.
Over the past two decades, great effort was expended in learning to manipulate and detect ferromagnetic order via spin-transfer torque (STT) 2,3 and spin pumping. 4 These two crucial processes enable the encoding and decoding of information in spin wave excitations. Such a program of using spin waves for information process-ing is termed magnonics. 12,13 In a magnonic computing scheme, spin waves are used in place of itinerant electrons for information transmission. As a result, no Joule heating is produced. In the magnetic insulator yttrium iron garnet, for instance, magnonic excitations can propagate coherently on centimeter scales. 12 Recently Kajiwara et al. have demonstrated empirically the transmission of information via STT and spin pumping processes in this material. 14 Do these same phenomena-namely, STT and spin pumping-exist in AFM materials? If so, it would illuminate a path toward fine-tuned detection and manipulation of these magnetic materials. There have been some papers discussing the interplay of spin current and AFM order, typically in metallic AFM cases. [15][16][17][18][19][20][21][22][23][24][25] Most recently, Cheng et al. demonstrated that the dynamics of antiferromagnetic Néel order can pump spin current from AFM to nonmagnetic interfaces at an efficiency comparable to the analogous ferromagnetic process. 26 Due to Onsager reciprocity, 27,28 the non-vanishing spin pumping implies the existence of a reciprocal STT acting on both the magnetic and Néel order in AFM. Consequently, it should be possible to excite antiferromagnetic spin wave modes by injecting spin current in a manner similar to spin wave excitation in ferromagnets.
In this article, we directly demonstrate the existence of this spin wave excitation. In doing so, we focus on AFM insulators; typical AFM materials such as MnF 2 or FeF 2 are insulators, and insulating properties also ensure the vanishing Joule heating which is so desirable for applications. [29][30][31] Several papers concerning AFM surface spin waves, [32][33][34] including waves in the presence of anisotropy and varied surface exchange energies, were published decades before the discovery of STT and spin pumping. By introducing STT at the AFM interface, we show that it is possible to excite AFM spin waves by spin current injection, and in particular to preferentially excite those surface spin wave modes promoted by certain surface anisotropies.
Collinear AFM order consists of two compensating magnetic sublattices. Due to the strong exchange coupling between the two sublattices, the spin wave modes in AFM typically have resonance frequencies in the terahertz regime. [32][33][34][35][36][37][38][39]  than the analogous excitations in ferromagnets, which operate in the gigahertz range. If the spin waves in AFM could be utilized for information processing, they may operate in this much higher THz range. Such devices could supersede those developed during our current era of GHz-speed information architecture. This paper is organized as follows: in Sec. II, we present a macrospin model which suggests not only that AFM spin waves can be excited by spin current, but also that the chirality of the spin wave depends on the spin current polarization. The former result means that information can be transmitted: one may imagine encoding binary data in the time-domain information of incoming spin waves. The latter result could allow us to robustly encode data in the spin wave chirality: for instance, a bit 0 is due to aẑ-polarized STT source while a bit 1 is due to a ẑ polarized STT. This distinction between source chiralities could markedly improve the fidelity of devices utilizing the magnetization domain for information processing. In Sec. III, we make the system more realistic by extending the dynamical equations from two sites to a full cubic lattice. Here we present results both for the bulk system and a semi-infinite system with an interface. Based on previous work, 40 we expect that surface e↵ects, by their role in modifying the magnons' activation threshold, will play an important part in the experimental realization of spin wave modes. In particular, we explore variations of the exchange coupling on the surfaces with both compensated and uncompensated net magnetizations. In Sec. IV, we o↵er concluding remarks and an application for experimental methods.

II. MACROSPIN MODEL
To provide a conceptual account of the mechanism underlying STT-generated spin waves in AF materials, we first present a minimal model which includes the impor-tant physical terms without the complications of a spatially extended lattice. The magnetization on the two sublattices are modeled as two macrospins, 41 m + and m , which are coupled by a constant Heisenberg-type exchange interaction ! J . They are additionally subject to Gilbert damping ↵ and spin transfer torque ! s ; the latter is due to an injected spin current polarized along theẑ direction. Both macrospins experience the a uniaxial anisotropy ! A in theẑ direction. 35 We also allow for an external magnetic field H 0ẑ along this axis. The setup is depicted schematically in Fig. 1, and yields an equation of motioṅ where ! H = H 0 . The e↵ective field term H ± e↵ is the negative derivative r m ± H of the Hamiltonian where the damping term is added phenomenologically, 39 and the STT term-where ! s is linear in the applied spin voltage-is derived in Appendix A.
In the small angle approximation, we demand that the deviation ✓ of m zẑ from m be small, so that m z = cos ✓ = 1 + O ✓ 2 and m x,y / ✓ + O ✓ 3 . Now theẑ-component of equation Eq. (1) vanishes to order O ✓ 2 , and the problem is reduced to two e↵ective dimensions in the xy-plane. We can exchange these two real dimensions for a single complex one by defining the transverse magnetization u ⌘ m x + im y and rewriting Eq. (1) in terms of this new variable. We then employ a spin wave ansatz u ± = µ ± e i!t which allows us to solve for the modes that satisfy equation Eq. (1). In the small-angle approximation, these eigenfrequencies of precession are The resonant frequency in the absence of damping and STT is ! 0 = p ! A (! A + 2! J ). In AFM, two degenerate modes with opposite chirality appear in Eq. (3). This equation highlights the essential competition between STT and precessional damping: when the applied spin current is su ciently strong, the second term in Eq. (3) becomes positive and selectively excites one of the ! ± modes depending on the sign of ! s . Therefore, spin waves with di↵erent chirality can be selectively excited according to the spin current polarization. This behavior is di↵erent from STT-induced FM dynamics, for which only one polarization of spin current can excite FM spin waves while the other polarization enhances damping instead.
We can re-express m ± in spherical angular coordinates (✓ ± , ' ± ) and derive a set of exact, coupled ODEs for this than the analogous excitations in ferromagnets, which operate in the gigahertz range. If the spin waves in AFM could be utilized for information processing, they may operate in this much higher THz range. Such devices could supersede those developed during our current era of GHz-speed information architecture.
This paper is organized as follows: in Sec. II, we present a macrospin model which suggests not only that AFM spin waves can be excited by spin current, but also that the chirality of the spin wave depends on the spin current polarization. The former result means that information can be transmitted: one may imagine encoding binary data in the time-domain information of incoming spin waves. The latter result could allow us to robustly encode data in the spin wave chirality: for instance, a bit 0 is due to aẑ-polarized STT source while a bit 1 is due to a −ẑ polarized STT. This distinction between source chiralities could markedly improve the fidelity of devices utilizing the magnetization domain for information processing. In Sec. III, we make the system more realistic by extending the dynamical equations from two sites to a full cubic lattice. Here we present results both for the bulk system and a semi-infinite system with an interface. Based on previous work, 40 we expect that surface effects, by their role in modifying the magnons' activation threshold, will play an important part in the experimental realization of spin wave modes. In particular, we explore variations of the exchange coupling on the surfaces with both compensated and uncompensated net magnetizations. In Sec. IV, we offer concluding remarks and an application for experimental methods.

II. MACROSPIN MODEL
To provide a conceptual account of the mechanism underlying STT-generated spin waves in AF materials, we first present a minimal model which includes the impor-tant physical terms without the complications of a spatially extended lattice. The magnetization on the two sublattices are modeled as two macrospins, 41 m + and m − , which are coupled by a constant Heisenberg-type exchange interaction ω J . They are additionally subject to Gilbert damping α and spin transfer torque ω s ; the latter is due to an injected spin current polarized along theẑ direction. Both macrospins experience the a uniaxial anisotropy ω A in theẑ direction. 35 We also allow for an external magnetic field H 0ẑ along this axis. The setup is depicted schematically in Fig. 1, and yields an equation of motioṅ where ω H = γH 0 . The effective field term H ± eff is the negative derivative −∇ m± H of the Hamiltonian where the damping term is added phenomenologically, 39 and the STT term-where ω s is linear in the applied spin voltage-is derived in Appendix A.
In the small angle approximation, we demand that the deviation θ of m zẑ from m be small, so that m z = cos θ = 1 + O θ 2 and m x,y ∝ θ + O θ 3 . Now theẑ-component of equation Eq. (1) vanishes to order O θ 2 , and the problem is reduced to two effective dimensions in the xy-plane. We can exchange these two real dimensions for a single complex one by defining the transverse magnetization u ≡ m x + im y and rewriting Eq. (1) in terms of this new variable. We then employ a spin wave ansatz u ± = µ ± e −iωt which allows us to solve for the modes that satisfy equation Eq. (1). In the small-angle approximation, these eigenfrequencies of precession are The resonant frequency in the absence of damping and STT is ω 0 = ω A (ω A + 2ω J ). In AFM, two degenerate modes with opposite chirality appear in Eq. (3). This equation highlights the essential competition between STT and precessional damping: when the applied spin current is sufficiently strong, the second term in Eq. (3) becomes positive and selectively excites one of the ω ± modes depending on the sign of ω s . Therefore, spin waves with different chirality can be selectively excited according to the spin current polarization. This behavior is different from STT-induced FM dynamics, for which only one polarization of spin current can excite FM spin waves while the other polarization enhances damping instead.
We can re-express m ± in spherical angular coordinates (θ ± , ϕ ± ) and derive a set of exact, coupled ODEs for this  Fig. 1, is plotted for t > 100 after the system has neared its steady state oscillation. Right: with stronger ! s , the system undergoes a spin flop, in which both m ± tilt to the north hemisphere. For both figures, ! A /! J = 0. 6. system directly from the coupled LLG equations. ✓ ± is taken to be the polar angle between m ± andẑ, and ± is the corresponding azimuthal angle. We finḋ where ' = ' + ' . This result is analytically exact. Some numerical calculations for these ODEs are depicted in Fig. 2. Since the exchange energy is locally minimized where ' + ' = ' = ⇡, we expect' + =' . In the small angle approximation and neglecting✓ ± terms, this condition is satisfied when where # ± are the angles that m ± make with the ±ẑ axes.
Choosing ' = ⇡, as energetically expected, recovers the results from Ref. 35. Within the✓ ⇡ 0 approximation, there is no real solution for # + = # in the presence of easy-axis anisotropy.

III. LATTICE CALCULATION
To consider a more realistic system than that of Sec. II, we now extend the Heisenberg-type Hamiltonian (2) to a simple cubic lattice as in Ref. 32: where the subscripts i and j are lattice sites and the first sum is taken over nearest neighbors.
We will consider both g-and a-type antiferromagnets. These configurations are depicted in Fig. 3, where the AFM terminates at x = 0 with compensated (left) and uncompensated (right) surfaces. We take the lattice constant as a = 1 so that the wavevector is dimensionless. The thermodynamic derivation from Sec. II is repeated for the Hamiltonian in Eq. (6) to derive an e↵ective onsite magnetic field. Knowledge of this field determines the LLG equation forṁ j , namely: The exchange coe cients ! ij will be uniformly constant ! ij = ! J for the g-type system where all nearest neighbors are the same, though for the a-type system we will need to distinguish ! ij = ! ? < 0 and ! ij = ! k > 0 for the coupling between inter-and intra-plane (respectively AFM-like and FM-like) neighbors.
By assuming a small precession of m j about its easyaxis, theẑ-component of the LLG Eqs. (7) can be neglected to first order. We then rewrite the equation of motion in terms of the transverse magnetization u ± ⌘ m ±,x + im ±,y as in Sec. II. Translational symmetry in time and the yz-plane validates the plane wave ansatz where j is the layer index in thex-direction and q is the wave vector in yz-plane. We substitute this equality into the transverse magnetization equation. From now on we will use k to refer to a 3D wavevector and q will be k's restriction in the yz-plane.
With these modifications, the LLG Eq. (7) is rewritten as a recurrence relation among di↵erent layers with j = µ + j µ j T . The square matrices S, N + , and N can be computed directly from considering the coe cients in Eq. (7). For g-type, with ⌦ = 6! J +! A i↵! and ! (g) q = 2! J (cos q y + cos q z ). For a-type, with ⌦ q = 2! ? + ! A + ! q i↵! and ! (a) q = 2! k (2 cos q y cos q z ).  Fig. 1, is plotted for t > 100 after the system has neared its steady state oscillation. Right: with stronger ωs, the system undergoes a spin flop, in which both m± tilt to the north hemisphere. For both figures, ωA/ωJ = 0. 6. system directly from the coupled LLG equations. θ ± is taken to be the polar angle between m ± andẑ, and φ ± is the corresponding azimuthal angle. We finḋ where ∆ϕ = ϕ + − ϕ − . This result is analytically exact. Some numerical calculations for these ODEs are depicted in Fig. 2. Since the exchange energy is locally minimized where ϕ + − ϕ − = ∆ϕ = π, we expectφ + =φ − . In the small angle approximation and neglectingθ ± terms, this condition is satisfied when where ϑ ± are the angles that m ± make with the ±ẑ axes.
Choosing ∆ϕ = π, as energetically expected, recovers the results from Ref. 35. Within theθ ≈ 0 approximation, there is no real solution for ϑ + = ϑ − in the presence of easy-axis anisotropy.

III. LATTICE CALCULATION
To consider a more realistic system than that of Sec. II, we now extend the Heisenberg-type Hamiltonian (2) to a simple cubic lattice as in Ref. 32: where the subscripts i and j are lattice sites and the first sum is taken over nearest neighbors. We will consider both g-and a-type antiferromagnets. These configurations are depicted in Fig. 3, where the AFM terminates at x = 0 with compensated (left) and uncompensated (right) surfaces. We take the lattice constant as a = 1 so that the wavevector is dimensionless. The thermodynamic derivation from Sec. II is repeated for the Hamiltonian in Eq. (6) to derive an effective onsite magnetic field. Knowledge of this field determines the LLG equation forṁ j , namely: The exchange coefficients ω ij will be uniformly constant ω ij = ω J for the g-type system where all nearest neighbors are the same, though for the a-type system we will need to distinguish ω ij = ω ⊥ < 0 and ω ij = ω > 0 for the coupling between inter-and intra-plane (respectively AFM-like and FM-like) neighbors.
By assuming a small precession of m j about its easyaxis, theẑ-component of the LLG Eqs. (7) can be neglected to first order. We then rewrite the equation of motion in terms of the transverse magnetization u ± ≡ m ±,x + im ±,y as in Sec. II. Translational symmetry in time and the yz-plane validates the plane wave ansatz where j is the layer index in thex-direction and q is the wave vector in yz-plane. We substitute this equality into the transverse magnetization equation. From now on we will use k to refer to a 3D wavevector and q will be k's restriction in the yz-plane. With these modifications, the LLG Eq. (7) is rewritten as a recurrence relation among different layers The square matrices S, N + , and N − can be computed directly from considering the coefficients in Eq. (7). For g-type, with Ω = 6ω J +ω A −iαω and ω (g) q = 2ω J (cos q y + cos q z ). For a-type, N (a) with Ω q = 2ω ⊥ + ω A + ω q − iαω and ω (a) q = 2ω (2 − cos q y − cos q z ). 3. (Color online) 2D slices of the spin configurations for g-type (left) and a-type (right) AFM, interfacing with NM. For g-type, the neighboring spins in the bulk have exchange coupling !J , but have coupling ✏!J on the surface. For a-type, the intralayer exchange coupling is ! k in the bulk and ✏! k on the surface, while the interlayer exchange coupling is ! ? . In both cases, the far left column of spins is the x = j = 0 atomic surface layer which sits against a nonmagnetic interface. Unit cells are outlined in dashed box. Spin current Is is injected from NM and exerting a torque on the surface spins.

A. Bulk calculation
For an infinite bulk without surface, there is also translational symmetry in thex direction, so we may take a plane wave solution for the x-coordinate: j = (q)e i(kxj !t) . We can then find the eigenfrequencies of AFM spin waves. For g-type, with ! k = 2! J (cos k x + cos k y + cos k z ). For the a-type lattice, we have an infinite stack of alternating ferromagnetic sheets. We choose forx to be the direction normal to any given sheet. Then the bulk dispersion is One can verify that in the limit ! k = 0 we recover decoupled 1D AF chains with the expected dispersion ±2! ? |sin k x | in the simple isotropic case. 42 Likewise, the ! ? = 0 limit recovers a decoupled 2D ferromagnetic system, with dispersion ±2! k |cos k y + cos k z 2| in the same simple case. The spin wave eigenfunctions corresponding to Eqs. (12,13) are respectively and ' (a) The dispersion relations (12,13) enforce a constraint linking the irreducible representations of time (!) and space (k) translational symmetries. For any particular ! and q, there are only a finite number of k x values in its preimage under the eigenvalue Eqs. (12,13). In the next section, we will need to consider linear combinations of bulk solutions to satisfy the boundary condition. The dispersion relations above will allow us to consider only a small subset of all conceivable wavenumbers k x .

B. The semi-infinite case
We now introduce an interface by terminating the insulator along its (100) plane and replacing the space x < 0 with a nonmagnetic contact from which spin current can be injected. We will modify the equations of motion to allow for special conditions on the atomic surface layer at x = 0.
First, an enhanced damping term is inserted into the LLG equation by taking ↵ 7 ! ↵ + j±,0 , where is the enhanced damping parameter for the surface spins. The STT term ! s m j ⇥ (ẑ ⇥ m j ) j±,0 is likewise included on the atomic surface layer. Finally, as a form of surface anisotropy, we allow a modulation of the intralayer exchange coupling represented by the ratio . It is known that this type of surface anisotropy can induce surface spin wave modes in AFM. 32 The variation in the exchange energy at the interface of magnetic materials has been studied by many groups. For instance, numerical studies on NiO(100) interfaces have shown that, depending on the assumptions of the model, surface exchange energy can vary by at least 20% with some groups showing as much as a 50% variation 43 from the bulk coupling.
We can write new equations of motion for this semiinfinite system as: where, for g-and a-types, respectively: and x,y,z are the Pauli matrices.

A. Bulk calculation
For an infinite bulk without surface, there is also translational symmetry in thex direction, so we may take a plane wave solution for the x-coordinate: ψ j = φ(q)e i(kxj−ωt) . We can then find the eigenfrequencies of AFM spin waves. For g-type, with ω k = 2ω J (cos k x + cos k y + cos k z ). For the a-type lattice, we have an infinite stack of alternating ferromagnetic sheets. We choose forx to be the direction normal to any given sheet. Then the bulk dispersion is One can verify that in the limit ω = 0 we recover decoupled 1D AF chains with the expected dispersion ±2ω ⊥ |sin k x | in the simple isotropic case. 42 Likewise, the ω ⊥ = 0 limit recovers a decoupled 2D ferromagnetic system, with dispersion ±2ω |cos k y + cos k z − 2| in the same simple case. The spin wave eigenfunctions corresponding to Eqs. (12,13) are respectively ϕ (g) and ϕ (a) The dispersion relations (12,13) enforce a constraint linking the irreducible representations of time (ω) and space (k) translational symmetries. For any particular ω and q, there are only a finite number of k x values in its preimage under the eigenvalue Eqs. (12,13). In the next section, we will need to consider linear combinations of bulk solutions to satisfy the boundary condition. The dispersion relations above will allow us to consider only a small subset of all conceivable wavenumbers k x .

B. The semi-infinite case
We now introduce an interface by terminating the insulator along its (100) plane and replacing the space x < 0 with a nonmagnetic contact from which spin current can be injected. We will modify the equations of motion to allow for special conditions on the atomic surface layer at x = 0.
First, an enhanced damping term is inserted into the LLG equation by taking α → α + βδ j±,0 , where β is the enhanced damping parameter for the surface spins. The STT term ω s m j × (ẑ × m j ) δ j±,0 is likewise included on the atomic surface layer. Finally, as a form of surface anisotropy, we allow a modulation of the intralayer exchange coupling represented by the ratio ≡ ω surf J /ω bulk J (or ω surf /ω bulk ). It is known that this type of surface anisotropy can induce surface spin wave modes in AFM. 32 The variation in the exchange energy at the interface of magnetic materials has been studied by many groups. For instance, numerical studies on NiO(100) interfaces have shown that, depending on the assumptions of the model, surface exchange energy can vary by at least 20% with some groups showing as much as a 50% variation 43 from the bulk coupling. We can write new equations of motion for this semiinfinite system as: where, for g-and a-types, respectively: and σ x,y,z are the Pauli matrices. We now take the bulk eigenvectors ϕ ± in Eq. (14) for the g-type as a basis for general solutions to a semi- infinite lattice configuration. By using the bulk dispersion relations, ϕ ± can be rewritten in terms of a distinguished eigenvalue ω and trigonometric functions of k x as in Eq. (14). Recall from the conclusion of Section III A that for a particular value of ω = ω(q) the irreducible representation k x is restricted to the finite set of values k x ∈ ω −1 (g,a) (ω (q)). We will call these at most four values by k µ . Since the cosine function is even, we see that two of the k µ values are related by a sign change to the other two. As will become clear in Sections III B 1 and III B 2, we demand that (k x ) be positive so that surface solutions decay into the bulk. Then two allowed values of k µ remain, which we call k + and k − .
We can now consider solutions of the form where ϕ ± are the bulk eigenvectors corresponding to k ± , which are the only allowed wavenumbers k x in the preimage of the bulk ω.
1. g-type, with compensated surface With Eq. (17), the boundary condition Eq. (15a) for the compensated g-type system takes the form The exponentials e ik ± can be determined from solving the eigenvalue equations Eq. (12,13) for cos k x , employing the Pythagorean identity to expand Euler's formula, and demanding solutions (k x ) > 0 which decay into the bulk. Taken together with Eq. (12), this equation can be solved analytically for ω when α = β = ω s = 0. This unperturbed eigenfrequency is then used to calculate constant perturbations-namely the iαω and βω terms-so that equation Eq. (18) can be evaluated to leading order in the presence of damping and STT with straightforward modifications to its coefficients. The results in the complex eigenfrequencies ω = ω r + iω i are plotted in Fig. 4, wherein the bulk modes are plotted as the shaded area and the surface modes are plotted in colored curves. To the leftmost panel of Fig. 4 corresponds the real part of the eigenfrequency ω r −ω H (in units of ω J ), and the right three panels are the imaginary part ω i /αω r for three different cases: purely intrinsic damping, with neither spin pumping nor STT; both damping and spin pumping, but no STT; and both damping and STT, but no spin pumping.
The dispersion relations of ω r for the surface modes of this system are plotted in Fig. 4 over a spectrum of surface exchange ratios . These surface modes are the same as those calculated in Ref. 32. The spin wave profiles for the surface modes are presented in lower panels of the figure, which shows a positive correlation between surface localization and surface anisotropy. These figures also reveal that the surface modes are acoustic (optical) modes for < 1 ( > 1).
Beyond the dispersion relations, we are also interested in the dissipative behavior of various spin wave modes. Especially of interest are their behavior under the influence a STT due to spin current injection from the NM contact. The second panel of Fig. 4 shows ω i when there is only intrinsic damping included. In this case there is neither spin pumping or STT, and we plot both the bulk modes (shaded continuum) and the surface modes (colored curves) for different values of the surface anisotropy . With the additional NM contact at the surface, the spin pumping into NM from AFM increases the dissipation for the spin wave modes, as seen in the third panel in Fig. 4. This enhancement is visible for the surface modes, but not for the bulk modes. This is due to the infinite "mass" of the latter. The introduction of a spintransfer torque can dramatically decrease the damping of some surface spin waves, especially in the low-regime where surface anisotropy is strong. The low damping combined with low excitation energy makes these lowmodes particularly excitable due to strong surface localization. Strong enough ω s together with low (strong surface anisotropy) can cause sign changes in ω i and lead to AFM spin wave excitation, as in the last panel of Fig. 4. Furthermore, STT distinguishes the two spin wave chiralities by enhancing the damping of one while reducing the other. Precisely which chirality is excited depends on the spin current polarization, so that it is distinctly possible to selectively excite a particular chiral mode.
2. a-type, with uncompensated surface For the uncompensated surface in an a-type AFM insulator, there is effectively only one k x which satisfies both the bulk eigenfrequency equations and the reality condition (k x ) > 0 for any given ω. The reasoning follows: first, the orientation of the unit cell is necessarily different in the a-type system, so that the coupling to the next unit cell along the x-direction requires a factor of e 2ik rather than just e ik in the a-type analog to equation Eq. (17); second, solving equation Eq. (13) for k x gives a family of four solutions-namely k, −k, π + k, and π − k-but as we mentioned in Sec. III B, only one of k and −k will have a positive imaginary part, and they furthermore will each appear identical to their π-shifted partners when expressed in the form e 2ik . This simplifies the form of the boundary condition Eq. (15a), as well as the a-type analog of Eq. (18). A similar procedure to that employed in the previous section is used to solve the unperturbed and then perturbed versions of this equation.
The spin wave dispersion ω r for an a-type AFM is different from that for g-type AFM; this is evident in the left panel of Fig. 5. However, the surface anisotropy still induces surface spin wave modes. Typical surface mode profile are shown below the dispersion plots. In the absorption spectra (right three panels of Fig. 5), the spin pumping (third panel) enhances the dissipation for both chiralities, while STT reduces the dissipation for one chirality and enhances the other. These results coincide with the outcomes of Section III B 1 for the g-type configuration, again distinguishing spin wave chiralities and demonstrating that a nonzero ω s in the a-type system can cause a change in sign of the absorption spectrum, and can consequently excite spin wave modes.

IV. CONCLUSIONS
In this article, we have directly revealed the purely electrical generation of spin waves in antiferromagnetic materials by using a polarized spin current to apply a spin-transfer torque to the surface layer. In particular, we found that surface spin wave modes induced by surface anisotropy are particularly easy to excite. This is reminiscent of-though decidedly different in character from-recent results which showed that STT-generated surface spin waves in YIG were enhanced by surface anisotropy. 40 Our work also takes a first step toward developing new experimental techniques for investigating antiferromagnets. Because it is relatively straightforward to generate a spin current and measure spin waves, STT-based methods could provide a new tool for probing and controlling AF materials. In particular, parameters such as damping, anisotropy, or surface exchange coupling could be inferred by retrofitting experimental data to models like those we present here. Since this data would be obtained by purely electrical means via a polarized spin current, it could be considerably easier to collect than neutron scattering results. Such a method could be complementary to current experimental procedures. In Sec. II we presented an extended LLG equation of motion which included a spin-transfer torque term induced by aẑ-polarized spin current: τ ± = ω s m ± × (ẑ × m ± ). This form of term is plausible on the grounds of right-hand-rule gymnastics, but in this appendix we provide a more rigorous derivation of its physical content.
We begin from Eqs. (6) of Ref. 26, which provide the STT on the m (magnetization) and n (staggered) sublattices due to an applied spin voltage V s , where V is the volume of the system, a is the lattice constant, and G r is the real part of the spin mixing conductance for an NM|AFM interface; the corresponding imaginary part of G is several orders of magnitudes smaller 26 and consequently ignored. By definition, we have m ± = m ± n on the two sublattices from Sec. II. Thus τ ± = τ m ± τ n , and the use of Eqs. (A1,A2) gives Now, as in the main text, we take the spin voltage to be collinear with the easy-axisẑ: V s = V sẑ . Allowing n ≈ 2m + ≈ −2m − , we have and we define the relevant constant of proportionality as ω s = (a 3 V s /eV)G r , thus achieving the form exhibited in equation Eq. (1).