Spin-transfer torque induced spin waves in antiferromagnetic insulators

The generation, control, and detection of pure spin current carried by spin waves is the subject of magnonics, [1–3] an emerging subfield of spintronics [4–7]. Typically studied in magnetic insulators, magnonics represents a promising platform for next-generation energy-efficient device architectures because spin wave transmission does not induce Joule heating.

So far the research efforts in magnonics have focused on ferromagnetic or effectively ferromagnetic insulators [1–3]. On the other hand, antiferromagnetic insulators are abundant in nature. Compared to ferromagnets, antiferromagnets (AFM) present several technological advantages. Their lack of stray field simplifies the issue of packing multiple nanoscale magnonic devices in close proximity. The resonant frequency of spin waves in AFM is typically THz, making faster dynamics and previously unused communications frequencies more readily available. Finally, in AFMs with easy-axis anisotropy, the spin wave modes are doubly degenerate with opposite chirality, offering a new degree of freedom for information encoding and processing. These advantages, together with recent experimental progress [8–11], make AFMs an attractive candidate for magnonics [12–14].

To realize a magnonic circuit in an AFM, the first priority is understanding the generation of spin waves. Optical excitation methods exist [15], but one would generally prefer an electrical procedure. Recently, Cheng et al showed that the spin-transfer torque (STT) [16–19] can be used to excite magnon modes in easy-axis AFMs [20]; the critical current required a priori to excite bulk modes in a typical AFM, though, is too large to be realistically produced by electronic means. One possible way to counter this problem is to excite surface spin waves [21]. It has been shown in ferromagnets that surface anisotropy can significantly lower the spin wave frequency and hence the critical current needed to excite the spin waves [22]. Our present concern is addressing whether a similar reduction in the critical current may be found for surface waves in AFM, bringing electrical excitation down to a more realistic regime.

In this article, we address this possibility of lowering the effective excitation threshold for AFM spin waves by considering the surface spin wave modes of AFMs. Our prediction relies on the fact that surface atoms in certain AFMs can have an effective exchange energy significantly lower than the bulk value [22–24]. One would then expect that the resonant frequency of spin waves localized to these exchange-reduced atoms will have lower excitation thresholds and be easier to excite. We compute the surface spin wave spectra of antiferromagnetic insulators with both magnetically compensated and uncompensated surfaces, and show that certain surface modes are, as expected, lower in energy. We then include a contribution from STT due to a polarized spin current injected at the surface. We find that this STT is sufficient to excite the surface spin waves in low-surface-exchange systems, which demonstrates a step toward making AFM magnonics more easily realizable in experiment. We also verify that the sign of the STT determines the handedness of the chiral AFM spin wave, as predicted [20, 25].

This paper is organized as follows: in section 2, we briefly review the eigenstructure of spin waves in antiferromagnetic systems by appealing to a simple two-spin system. In section 3, we review solutions of the LLG equations on an infinite cubic AFM lattice, and then present new results on AFM spin wave spectra for a semi-infinite system with an interface. Based on previous work [22], 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 section 4, we offer concluding remarks and an application for experimental methods.

To provide a conceptual account of the mechanism underlying STT-generated spin waves in AFM materials, we present a minimal model which includes the important physical terms without the complications of a spatially extended lattice. This so-called 'macrospin model' has been considered by many in the past, and we include it here not as new work but as a pedagogical tool to illustrate the mechanism by which STT determines AFM spin wave handedness. This idea is extended to our core result on a semi-infinite lattice in section 3.

In the macrospin model, the magnetization on the two sublattices are modeled as two macrospins [26], m₊ and ${{\bf{m}}}_{-},$ 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 the $\hat{z}$ direction. Both macrospins experience a uniaxial anisotropy ω_A in the $\hat{z}$ direction [27]. We also allow for an external magnetic field ${H}_{0}\hat{z}$ along this axis. The setup yields an equation of motion

$$\begin{eqnarray}{\dot{{\bf{m}}}}_{\pm } & = & -{{\bf{m}}}_{\pm }\times {{\bf{H}}}_{\mathrm{eff}}^{\pm }+\alpha {{\bf{m}}}_{\pm }\times {\dot{{\bf{m}}}}_{\pm }+{\omega }_{{\rm{s}}}{{\bf{m}}}_{\pm }\times \left(\hat{z}\times {{\bf{m}}}_{\pm }\right)\\ & = & -{{\bf{m}}}_{\pm }\times \left[-{\omega }_{J}{{\bf{m}}}_{\mp }+\left({\omega }_{{\rm{A}}}{m}_{\pm ,z}+{\omega }_{H}\right)\hat{z}\right]\\ & & +\alpha {{\bf{m}}}_{\pm }\times {\dot{{\bf{m}}}}_{\pm }+{\omega }_{{\rm{s}}}{{\bf{m}}}_{\pm }\times \left(\hat{z}\times {{\bf{m}}}_{\pm }\right),\end{eqnarray} \tag{ 1 }$$

where ω_H = γ H₀. The effective field term ${{\bf{H}}}_{\mathrm{eff}}^{\pm }$ is the negative derivative $-{{\rm{\nabla }}}_{{{\bf{m}}}_{\pm }}H$ of the Hamiltonian

$$\begin{eqnarray}&&H={\omega }_{J}{{\bf{m}}}_{+}\cdot {{\bf{m}}}_{-}-{\omega }_{H}\left({m}_{+}^{z}+{m}_{-}^{z}\right)-\displaystyle \frac{{\omega }_{{\rm{A}}}}{2}\left({{m}_{+}^{z}}^{2}+{{m}_{-}^{z}}^{2}\right)\end{eqnarray} \tag{ 2 }$$

where the damping term is added phenomenologically [28], and the STT term—where ω_s is linear in the applied spin voltage—is due to [20]; we partially rederive it for the reader in appendix A.

In the small angle approximation, we demand that the deviation θ of ${m}_{z}\hat{z}$ from ${\bf{m}}$ be small, so that ${m}_{z}=\mathrm{cos}\theta =1+O\left({\theta }^{2}\right)$ and ${m}_{x,y}\propto \theta +O\left({\theta }^{3}\right).$ Now the $\hat{z}$-component of equation (1) vanishes to order O(θ²), 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 equation (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 (1). In the small-angle approximation, these eigenfrequencies of precession are

$$\begin{eqnarray}&&{\omega }_{\pm }=\pm {\omega }_{0}-{\rm{i}}\alpha \left({\omega }_{J}+{\omega }_{{\rm{A}}}\right)\left(1\mp \displaystyle \frac{{\omega }_{{\rm{s}}}}{\alpha {\omega }_{0}}\right).\end{eqnarray} \tag{ 3 }$$

The resonant frequency in the absence of damping and STT is ${\omega }_{0}=\sqrt{{\omega }_{{\rm{A}}}\left({\omega }_{{\rm{A}}}+2{\omega }_{J}\right)}.$ In AFM, two degenerate modes with opposite chirality appear in equation (3). This equation highlights the essential competition between STT and precessional damping: when the applied spin current is sufficiently strong, the dissipative term (second term) in equation (3) changes sign 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 since FM magnons can only accept spin antiparallel to the local magnization. We plot a solution of these equations in figure 1. We see on the left of the figure that pumping a positive ${\hat{S}}_{z}$ current into the system via the STT mechanism described above preferentially selects a mode with positive net magnetization ${m}_{z}={m}_{z}^{+}+{m}_{z}^{-}\gt 0.$

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In appendix B, we carry this derivation a bit further to demonstrate explicitly the degree of magnetization carried by the macrospin model in one of its chiral eigenstates. The interested reader may refer there for more detail.

To consider a more realistic system than that of section 2, we now extend the Heisenberg-type Hamiltonian (2) to a simple cubic lattice as in [29]:

$$\begin{eqnarray}&&H=\displaystyle \sum _{\langle i,j\rangle}{\omega }_{{ij}}{{\bf{m}}}_{i}\cdot {{\bf{m}}}_{j}-\displaystyle \sum _{j}\left({\omega }_{H}+\displaystyle \frac{{\omega }_{{\rm{A}}}}{2}{m}_{j,z}\right){m}_{j,z},\end{eqnarray} \tag{ 4 }$$

where the subscripts i and j are lattice sites and the first sum is taken over nearest neighbors.

We are interested in both magnetically compensated (left) and uncompensated (right) surfaces, as depicted in figure 2. We take g- and a-type AFM as prototypical textures for these two cases. g-type AFMs are widely studied; some examples are La₂CuO₄ [30, 31] and BiFeO₃ [32]. On the other hand, there are quite a few a-type AFMs, mostly in layered magnetic compounds such as RbMnCl₃ [33], and certain transition metal trichalcogenides [34]. Even for AFMs with cubic symmetry, depending on the cut angle, the surface can be uncompensated as well, such as (111) NiO [35] or (111) BiFeO₃.

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We take the lattice constant as a = 1 so that the wavevector is dimensionless. The derivation from section 2 is repeated for the Hamiltonian in equation (4) to derive an effective on-site magnetic field. Knowledge of this field determines the LLG equation for ${\dot{{\bf{m}}}}_{j},$ namely:

$$\begin{eqnarray}&&{\dot{{\bf{m}}}}_{j}=-{{\bf{m}}}_{j}\times \left[\displaystyle \sum _{\langle i,j\rangle}{\omega }_{{ij}}{{\bf{m}}}_{i}-({\omega }_{H}-{\omega }_{{\rm{A}}}{m}_{j,z})\hat{z}\right]+\alpha {{\bf{m}}}_{j}\times {\dot{{\bf{m}}}}_{j}.\end{eqnarray} \tag{ 5 }$$

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 ${\omega }_{{ij}}={\omega }_{\parallel }\gt 0$ for the coupling between inter- and intra-plane (respectively AFM-like and FM-like) neighbors.

By assuming a small precession of ${{\bf{m}}}_{j}$ about its easy-axis, the $\hat{z}$-component of the LLG equation (5) can be neglected to first order. We then rewrite the equation of motion in terms of the transverse magnetization ${u}^{\pm }\equiv {m}_{\pm ,x}+{\rm{i}}{m}_{\pm ,y}$ as in section 2. Translational symmetry in time and the yz-plane validates the plane wave ansatz

$$\begin{eqnarray}&&{u}_{(j,{\bf{s}})}^{\pm }={\mu }_{j,{\bf{q}}}^{\pm }{{\rm{e}}}^{{\rm{i}}\left({\bf{q}}\cdot {\bf{s}}-\omega t\right),}\end{eqnarray} \tag{ 6 }$$

where j is the layer index in the $\hat{x}$-direction and ${\bf{q}}$ is the wave vector in yz-plane. We substitute this equality into the transverse magnetization equation. From now on we will use ${\bf{k}}$ to refer to a 3D wavevector and ${\bf{q}}$ will be ${\bf{k}}$'s restriction in the yz-plane.

With these modifications, the LLG equation (5) is rewritten as a recurrence relation among different layers

$$\begin{eqnarray}&&S{\psi }_{j}+{N}_{+}{\psi }_{j+1}+{N}_{-}{\psi }_{j-1}=0\end{eqnarray} \tag{ 7 }$$

with ${\psi }_{j}={\left({\mu }_{j}^{+}{\mu }_{j}^{-}\right)}^{T}.$ The square matrices S, N₊, and N₋ can be computed directly from considering the coefficients in equation (5). For g-type,

$$\begin{eqnarray}{S}^{(g)}=\left(\begin{array}{lc}\omega -{\omega }_{H}-{\rm{\Omega }} & -{\omega }_{q}^{(g)}\\ {\omega }_{q}^{(g)} & \omega -{\omega }_{H}+{\rm{\Omega }}\end{array}\right),\end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray}{N}_{+}^{(g)}={N}_{-}^{(g)}=\left(\begin{array}{cc}0 & -{\omega }_{J}\\ {\omega }_{J} & 0\end{array}\right)\end{eqnarray} \tag{ 8b }$$

with Ω = 6ω_J + ω_A − iαω and ${\omega }_{q}^{(g)}=2{\omega }_{J}\left(\mathrm{cos}{q}_{y}+\mathrm{cos}{q}_{z}\right).$ For a-type,

$$\begin{eqnarray}{S}^{(a)}=\left(\begin{array}{lc}\omega -{\omega }_{H}-{{\rm{\Omega }}}_{q} & -{\omega }_{\perp }\\ {\omega }_{\perp } & \omega -{\omega }_{H}+{{\rm{\Omega }}}_{q}\end{array}\right),\end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray}{N}_{+}^{(a)}=\left(\begin{array}{lc}0 & -{\omega }_{\perp }\\ 0 & 0\end{array}\right),{N}_{-}^{(a)}=\left(\begin{array}{ll}0 & 0\\ {\omega }_{\perp } & 0\end{array}\right)\end{eqnarray} \tag{ 9b }$$

with ${{\rm{\Omega }}}_{q}=2{\omega }_{\perp }+{\omega }_{{\rm{A}}}+{\omega }_{q}^{(a)}-{\rm{i}}\alpha \omega$ and ${\omega }_{q}^{(a)}=2{\omega }_{\parallel }\left(2-\mathrm{cos}{q}_{y}-\mathrm{cos}{q}_{z}\right).$

3.1. Bulk calculation

For comparison with our results for a semi-infinite lattice in section 3.2, we pause to reproduce the bulk spin wave spectrum this formalism. The reader may refer to [29], or to any condensed matter theory textbook, for a more complete discussion of the g-type spectral calculation. The a-type calculation is similar except that the primitive lattice vectors differ.

In addition to the translational symmetries used in the previous section, a bulk lattice possesses an additional translational symmetry in the $\hat{x}$ direction. Therefore we may take a plane wave solution for the x-coordinate: ${\psi }_{j}=\phi (q){{\rm{e}}}^{{\rm{i}}({k}_{x}j-\omega t)}.$ We can then find the eigenfrequencies of AFM spin waves. For g-type,

$$\begin{eqnarray}&&{\omega }_{(g)}={\omega }_{H}\pm \sqrt{{{\rm{\Omega }}}^{2}-{\omega }_{k}^{2}}\end{eqnarray} \tag{ 10 }$$

with ${\omega }_{k}=2{\omega }_{J}\left(\mathrm{cos}{k}_{x}+\mathrm{cos}{k}_{y}+\mathrm{cos}{k}_{z}\right).$ For the a-type lattice, we have an infinite stack of alternating ferromagnetic sheets. We choose for $\hat{x}$ to be the direction normal to any given sheet. Then the bulk dispersion is

$$\begin{eqnarray}&&{\omega }_{(a)}={\omega }_{H}\pm \sqrt{{{\rm{\Omega }}}_{q}^{2}-4{\omega }_{\perp }^{2}{\mathrm{cos}}^{2}{k}_{x}}.\end{eqnarray} \tag{ 11 }$$

One can verify that in the limit ω_∥ = 0 we recover decoupled 1D AF chains with the expected dispersion $\pm 2{\omega }_{\perp }\left|\mathrm{sin}{k}_{x}\right|$ in the simple isotropic case [36]. Likewise, the ω_⊥ = 0 limit recovers a decoupled 2D ferromagnetic system, with dispersion $\pm 2{\omega }_{\parallel }\left|\mathrm{cos}{k}_{y}+\mathrm{cos}{k}_{z}-2\right|$ in the same simple case.

The spin wave eigenfunctions corresponding to equations (10) and (11) are respectively

$$\begin{eqnarray}&&{\varphi }_{\pm }^{\left(g\right)}=\left(\begin{array}{l}\sqrt{{\rm{\Omega }}+\left(\omega -{\omega }_{H}\right)}\\ \mp \sqrt{{\rm{\Omega }}-\left(\omega -{\omega }_{H}\right)}\end{array}\right),\end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray}&&\mathrm{and}\ {\varphi }_{\pm }^{\left(a\right)}=\left(\begin{array}{l}{{\rm{e}}}^{{\rm{i}}{k}_{x}}\sqrt{{{\rm{\Omega }}}_{q}\pm \left(\omega -{\omega }_{H}\right)}\\ -\sqrt{{{\rm{\Omega }}}_{q}\mp \left(\omega -{\omega }_{H}\right)}\end{array}\right).\end{eqnarray} \tag{ 12b }$$

The dispersion relations (10) and (11) enforce a constraint linking the irreducible representations of time (ω) and space (k) translational symmetries. For any particular ω and ${\bf{q}},$ there are only a finite number of k_x values in its preimage under the eigenvalue equations (10) and (11). 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.

3.2. 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 non-magnetic contact from which spin current can be injected; see figure 2. 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 [37] is inserted into the LLG equation by taking α ↦ α + βδ_{j ±,0}, where β is the enhanced damping parameter for the surface spins. This enhanced damping represents spin loss due to spin pumping from the AFM back into the NM contact. The STT term ${\omega }_{{\rm{s}}}{{\bf{m}}}_{j}\times \left(\hat{{\bf{z}}}\times {{\bf{m}}}_{j}\right){\delta }_{j\pm ,0}$ is likewise included on the atomic surface layer. This is an anti-damping term; formally, there is also a field-like term. This field-like term, however, is proportional to the imaginary part of the spin mixing conductance, which is several orders of magnitude smaller than the real part which gives the scale of the anti-damping contribution [20]. Therefore, we keep only the anti-damping term in this calculuation. Finally, as a form of surface anisotropy, we allow a modulation of the intralayer exchange coupling represented by the ratio $\epsilon \equiv {\omega }_{J}^{\mathrm{surf}}/{\omega }_{J}^{\mathrm{bulk}}$ (or ${\omega }_{\parallel }^{\mathrm{surf}}/{\omega }_{\parallel }^{\mathrm{bulk}}$). It is known that this type of surface anisotropy can modulate the surface spin wave modes in AFM [29]. 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 [23] from the bulk coupling.

We can write new equations of motion for this semi-infinite system as:

$$\begin{eqnarray}&&\left(S+B\right){\psi }_{0}+{N}_{+}{\psi }_{1}=0\quad \left(,j=0\right),\end{eqnarray} \tag{ 13a }$$

$$\begin{eqnarray}&&S{\psi }_{j}+{N}_{+}{\psi }_{j+1}+{N}_{-}{\psi }_{j-1}=0\quad \left(,j\gt 0\right),\end{eqnarray} \tag{ 13b }$$

where, for g- and a-types, respectively:

$$\begin{eqnarray}&&{B}^{(g)}=\left[\left(5-4\epsilon \right){\omega }_{J}+{\rm{i}}\left({\omega }_{{\rm{s}}}+\beta \omega \right)\right]{\sigma }_{z}+{\omega }_{q}\left(1-\epsilon \right){\rm{i}}{\sigma }_{y},\end{eqnarray} \tag{ 14a }$$

$$\begin{eqnarray}&&{B}^{(a)}=-\left[{\omega }_{\perp }+{\omega }_{q}\left(1-\epsilon \right)-{\rm{i}}\left({\omega }_{s}-\beta \omega \right)\right]\displaystyle \frac{1-{\sigma }_{z}}{2},\end{eqnarray} \tag{ 14b }$$

and σ_{x, y, z} are the Pauli matrices.

We now take the bulk eigenvectors φ_± in equation (12) 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 equation (12). Recall from the conclusion of section 3.1 that for a particular value of ω = ω(q) the irreducible representation k_x is restricted to the finite set of values ${k}_{x}\in {\omega }_{(g,a)}^{-1}(\omega \left({\bf{q}}\right)).$ 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 3.2.1 and 3.2.2, we demand that ${\mathfrak{I}}\left({k}_{x}\right)$ 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

$$\begin{eqnarray}&&{\psi }_{j}={\eta }_{+}{\varphi }_{+}{{\rm{e}}}^{{\rm{i}}({k}^{+}j-\omega t)}+{\eta }_{-}{\varphi }_{-}{{\rm{e}}}^{{\rm{i}}({k}^{-}j-\omega t),}\end{eqnarray} \tag{ 15 }$$

where φ_± are the bulk eigenvectors corresponding to k^±, which are the only allowed wavenumbers k_x in the preimage of the bulk ω.

3.2.1. g-type, with compensated surface

Using equation (15), the boundary condition equation (13a) for the compensated g-type system takes the form

$$\begin{eqnarray}&&\mathrm{det}\left[B\left({\varphi }_{+}\ {\varphi }_{-}\right)+N\left({\varphi }_{+}{{\rm{e}}}^{{\rm{i}}{k}^{+}}\;{\varphi }_{-}{{\rm{e}}}^{{\rm{i}}{k}^{-}}\right)\right]=0.\end{eqnarray} \tag{ 16 }$$

The exponentials ${{\rm{e}}}^{{\rm{i}}{k}^{\pm }}$ can be determined from solving the eigenvalue equations equation (10), (11) for $\mathrm{cos}{k}_{x},$ employing the Pythagorean identity to expand Euler's formula, and demanding solutions ${\mathfrak{I}}\left({k}_{x}\right)\gt 0$ which decay into the bulk. Taken together with equation (10), 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 (16) 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 figure 3, wherein the bulk modes are plotted as the shaded area and the surface modes are plotted in colored curves. To the leftmost panel of figure 3 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 (due to the enhanced damping, β), but no STT; and finally damping, STT, and spin pumping all together.

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The dispersion relations of ω_r for the surface modes of this system are plotted in figure 3 over a spectrum of surface exchange ratios . These surface modes are the same as those calculated in [29]. The spin wave profiles for the surface modes are presented in lower panels of the figure. These figures also reveal that the surface modes in a g-type AFM can be either acoustic or optical; a detailed discussion of the acoustic/optical transition as a function of is given in [29].

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 figure 3 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 surface spin wave modes, as seen in the third panel in figure 3. Far from the γ point, the effective damping enhancement (in the language of [38]) is Δα ≈ β. This is expected since this regime corresponds to high surface localization, wherein β is effectively just added to α in the local LLG equations. Introduction of a STT can dramatically decrease the damping of some surface spin waves, especially in the low- regime. The low damping combined with low excitation energy makes these low- modes particularly excitable due to strong surface localization. Strong enough ω_s together with low (weakened surface exchange coupling) can cause sign changes in ω_i and lead to AFM spin wave excitation, as in the last panel of figure 3. 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.

3.2.2. 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 ${\mathfrak{I}}\left({k}_{x}\right)\gt 0$ 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 equation (15); second, solving equation (11) for k_x gives a family of four solutions—namely k − k, π + k, and π−k—but as we mentioned in section 3.2, 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 equation (13a), as well as the a-type analog of equation (16). A procedure similar 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 figure 4. However, the surface anisotropy still modifies the surface spin wave modes. Typical surface mode profiles are shown below the dispersion plots. In the absorption spectra (right three panels of figure 4), the spin pumping (third panel) enhances the dissipation for both chiralities (again at Δα ≈ β) while STT reduces the dissipation for one chirality and enhances the other. These results coincide with the outcomes of section 3.2.1 for the g-type configuration, again distinguishing spin wave chiralities and demonstrating that a non-zero ω_s in the a-type system can cause a change in sign of the absorption spectrum, and can consequently excite spin wave modes.

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In this article, we have calculated the spin wave spectrum of STT-induced AFM surface excitations. In particular, we found that surface spin wave modes modulated by surface anisotropy are particularly easy to excite compared to bulk modes, implying a lowering of the critical current required to perform magnonic operations in AFM insulators.

The efficiency of spin pumping processes in AFMs is known to be comparable to their ferromagnetic cousins [20]. However, because AFMs have a much stronger exchange coupling, an a priori estimate of the threshold current for exciting AFM surface spin waves is two to three orders of magnitude higher than in ferromagnet insulators. Nevertheless, the critical current for exciting a ferromagnetic magnon current was found in [39] to be two to three orders of magnitude lower than the expectation initially accorded to YIG's resonant frequency. If the same unforeseen reduction occurs in AFM, then the critical current would be on the order J_c ≈ 10⁸A cm⁻², which is within experimental feasibility. Our contribution is to take a first step in investigating this potential reduction in the critical barrier. One may of course seek materials with appropriate exchange or anisotropy energies in accordance with equation (3) in order to lower the barrier; we find that seeking materials with low surface exchange coupling reduces the threshold further.

Our work also takes a first step toward developing new experimental techniques for investigating AFM. 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 a powerful complement to current experimental procedures, but is intractable without an understanding of the spin wave response to surface STT akin to that which we have outlined above. In any case, such a scheme would require considerable refinement to what we have presented in this article; one would want to keep higher order terms, introduce another thin-film boundary, and break translational invariance along the surface. Treating non-single-crystal AFMs would introduce even more complication. We leave these details to future research, noting here only that continual improvement of our understanding of AFM spin waves should begin to open new routes to experimental investigation on the topic.

This work was supported by the National Science Foundation, Office of Emerging Frontiers in Research and Innovation EFRI-1433496 (MWD and DX), the U S Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division (GMS ), and by the special funds for the Major State Basic Research Project of China (grants No. 2014CB921600, No. 2011CB925601) and the National Natural Science Foundation of China (grant No. 91121002) (WG and JX).

In section 2 we presented an extended LLG equation of motion which included a STT term induced by a $\hat{z}$-polarized spin current: ${\tau }_{\pm }={\omega }_{{\rm{s}}}{{\bf{m}}}_{\pm }\times \left(\hat{z}\times {{\bf{m}}}_{\pm }\right).$ This form of term is plausible phenomenologically, but in this appendix we provide a more rigorous derivation of its physical content.

We begin from equation (6) of [20], which provide the STT on the ${\bf{m}}$ (magnetization) and ${\bf{n}}$ (staggered) sublattices due to an applied spin voltage ${{\bf{V}}}_{{\rm{s}}}$,

$$\begin{eqnarray}&&{\tau }_{{\rm{n}}}=-\displaystyle \frac{{a}^{3}}{{\rm{e}}{\mathcal{V}}}{G}_{{\rm{r}}}{\bf{n}}\times \left({\bf{m}}\times {{\bf{V}}}_{{\rm{s}}}\right)\end{eqnarray} \tag{ A.1 }$$

$$\begin{eqnarray}&&{\tau }_{{\rm{m}}}=-\displaystyle \frac{{a}^{3}}{{\rm{e}}{\mathcal{V}}}{G}_{{\rm{r}}}{\bf{n}}\times \left({\bf{n}}\times {{\bf{V}}}_{{\rm{s}}}\right),\end{eqnarray} \tag{ A.2 }$$

where ${\mathcal{V}}$ 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 [20] and consequently ignored. By definition, we have ${{\bf{m}}}_{\pm }={\bf{m}}\pm {\bf{n}}$ on the two sublattices from section 2. Thus τ_± = τ_m ± τ_n, and the use of equations (A.1), (A.2) gives

$$\begin{eqnarray}&&{\tau }_{\pm }=-\displaystyle \frac{{a}^{3}}{{\rm{e}}{\mathcal{V}}}{\bf{n}}\times \left(\pm {G}_{{\rm{r}}}{{\bf{m}}}_{\pm }\times {{\bf{V}}}_{{\rm{s}}}\right).\end{eqnarray} \tag{ A.3 }$$

Now, as in the main text, we take the spin voltage to be collinear with the easy-axis $\hat{z}:$ ${{\bf{V}}}_{{\rm{s}}}={V}_{{\rm{s}}}\hat{z}.$ Allowing ${\bf{n}}\approx 2{{\bf{m}}}_{+}\approx -2{{\bf{m}}}_{-},$ we have

$$\begin{eqnarray}&&{\tau }_{\pm }=\displaystyle \frac{{a}^{3}{V}_{{\rm{s}}}}{{\rm{e}}{\mathcal{V}}}{G}_{{\rm{r}}}{{\bf{m}}}_{\pm }\times \left(\hat{z}\times {{\bf{m}}}_{\pm }\right),\end{eqnarray} \tag{ A.4 }$$

and we define the relevant constant of proportionality as ${\omega }_{{\rm{s}}}=({a}^{3}{V}_{{\rm{s}}}/{\rm{e}}{\mathcal{V}}){G}_{{\rm{r}}},$ thus achieving the form exhibited in equation (1).

In section 2, we solved a two-spin system coupled by antiferromagnetic exchange and including relevant terms (anisotropy, STT, etc.) that were needed in the main text. Here, we push that derivation a bit further to provide the reader with some more qualitative details concerning the chiral eigenmodes that emerge from this prototype system.

We can re-express ${{\bf{m}}}_{\pm }$ in spherical angular coordinates (θ^±, φ^±) and derive a set of exact, coupled ODEs for this system directly from the coupled LLG equations of section 2. θ^± is taken to be the polar angle between ${{\bf{m}}}_{\pm }$ and $\hat{z},$ and ϕ^± is the corresponding azimuthal angle. We find

$$\begin{eqnarray}&&{\dot{\theta }}^{+}={\omega }_{J}\mathrm{sin}{\rm{\Delta }}\varphi \mathrm{sin}{\theta }^{-}-\left(\alpha {\dot{\varphi }}^{+}+{\omega }_{{\rm{s}}}\right)\mathrm{sin}{\theta }^{+},\end{eqnarray} \tag{ B1a }$$

$$\begin{eqnarray}{\dot{\varphi }}^{+} & = & {\omega }_{H}+{\omega }_{{\rm{A}}}+{\omega }_{J}\mathrm{sin}{\theta }^{-}\mathrm{cot}{\theta }^{+}\mathrm{cos}{\rm{\Delta }}\varphi \\ & & -{\omega }_{J}\mathrm{cos}{\theta }^{-}+\alpha {\dot{\theta }}^{+}\mathrm{csc}{\theta }^{+},\end{eqnarray} \tag{ B1b }$$

$$\begin{eqnarray}&&{\dot{\theta }}^{-}=-{\omega }_{J}\mathrm{sin}{\rm{\Delta }}\varphi \mathrm{sin}{\theta }^{+}-\left(\alpha {\dot{\varphi }}^{-}+{\omega }_{{\rm{s}}}\right)\mathrm{sin}{\theta }^{-},\end{eqnarray} \tag{ B1c }$$

$$\begin{eqnarray}{\dot{\varphi }}^{-} & = & {\omega }_{H}-{\omega }_{A}+{\omega }_{J}\mathrm{sin}{\theta }^{+}\mathrm{cot}{\theta }^{-}\mathrm{cos}{\rm{\Delta }}\varphi \\ & & -{\omega }_{J}\mathrm{cos}{\theta }^{+}+\alpha {\dot{\theta }}^{-}\mathrm{csc}{\theta }^{-},\end{eqnarray} \tag{ B1d }$$

where Δφ = φ⁺ − φ⁻. This result is analytically exact. Some numerical calculations for these ODEs are depicted in figure 1. Since the exchange energy is locally minimized where φ⁺ − φ⁻ = Δφ = π, we expect ${\dot{\varphi }}^{+}\;=\;{\dot{\varphi }}^{-}.$ In the small angle approximation and neglecting ${\dot{\theta }}_{\pm }$ terms, this condition is satisfied when

$$\begin{eqnarray}&&\displaystyle \frac{{\vartheta }^{+}}{{\vartheta }^{-}}=-\left(\displaystyle \frac{{\omega }_{J}+{\omega }_{A}}{{\omega }_{J}\mathrm{cos}{\rm{\Delta }}\varphi }\right)\pm \sqrt{{\left(\displaystyle \frac{{\omega }_{J}+{\omega }_{A}}{{\omega }_{J}\mathrm{cos}{\rm{\Delta }}\varphi }\right)}^{2}-1}.\end{eqnarray} \tag{ B2 }$$

where ϑ^±are the angles that ${{\bf{m}}}_{\pm }$ make with the $\pm \hat{z}$ axes. Choosing Δφ = π, as energetically expected, recovers the results from [27]. Within the $\dot{\theta }\approx 0$ approximation, there is no real solution for ϑ⁺ = ϑ⁻ in the presence of easy-axis anisotropy, and one spin will always dominate the dynamics. Because the spins stay antiparallel, the two chiral modes correspond to a right-handed or left-handed rotation of the (+)-sublattice, and always carry a net angular momentum [20].

The fact that these modes carry a non-zero packet of spin is critical for both the experimental detection and technological application of antiferromagnetic magnonics. It means that these spin waves carry a spin current, even though the two AFM sublatices would naïvely cancel each other's magnetizations. The association of opposite spin to each chiral mode is what gives rise to the chiral selection mechanism of figures 3 and 4: pumping opposite spin current into the system (via STT) will excite opposite chiral modes.