Emission of fast-propagating spin waves by an antiferromagnetic domain wall driven by spin current

Antiferromagnets (AFMs) have great benefits for spintronic applications such as high frequencies (up to THz), high speeds (up to tens of km/s) of magnetic excitations, and field-free operation. Advanced devices will require high-speed propagating spin waves (SWs) as signal carriers, i.e., SWs with high k-vectors, the excitation of which remains challenging. We show that a domain wall (DW) in anisotropic AFM driven by the spin current can be a source of such propagating SWs with high frequencies and group velocities. In the proposed generator, the spin current, with polarization directed along the easy anisotropy axis, excites the precession of the N\'eel vector within the DW. The threshold current is defined by the value of the anisotropy in the hard plane, and the frequency of the DW precession is tuneable by the strength of the spin current. We show that the above precession of spins inside the DW leads to robust emission of high-frequency propagating SWs into the AFM strip with very short wavelengths comparable to the exchange length, which is hard to achieve by any other method.

Introduction.-Spin-transfer-torque and spin-Hall auto-oscillators (AOs) based on ferromagnetic materials (FMs) are well-established devices in modern spintronics and have a great potential for advanced signal and data processing [1][2][3][4].For example, thanks to their highly nonlinear behavior, they are promising in neuromorphic computing applications, such as image or sound recognition [5][6][7].Such complex tasks require large arrays of strongly mutually coupled AOs that can be achieved by direct exchange, magneto-dipolar interactions, or spin waves (SWs) propagating between individual AOs [8][9][10][11][12][13].The latter has special advantages since SWs can carry signals on large distances and be additionally processed in the inter-AOs space [7,14,15].Despite the above benefits, the FM AOs have significant drawbacks, such as the necessity of externally applied strong magnetic field, low operational frequencies, which are usually limited by a few tens of GHz [16], and low velocity of the emitted SWs, which are of the order of 1 km/s.
Recently, it was proposed to use antiferromagnetic materials (AFM) instead of FM to eliminate the above issues [17][18][19][20][21][22][23][24][25].AFM AOs can operate in the THz frequency range and do not require an external magnetic field due to the well-known feature of the AFM spin dynamics -the utilization of the internal exchange field or so-called -exchange amplification [26][27][28].The velocities of the SWs in AFMs can reach dozens of km/s [29], which is promising for the fast signal/data transduction between AOs.However, substantially short wavelengths of the excited magnons are required to achieve such high velocities.The dispersion relation for the propagating SWs in AFM reads as ω = ω 2 0 + c 2 k 2 (where ω 0 is the frequency of AFM resonance (AFMR), c -maximum group velocity of magnons and k denotes a wavevector).Thus, the group SW velocity, v gr = ∂ω/∂k, tends to zero for a small k, and one is interested in the case k ≳ ω 0 /c, which corresponds to the wavelength of a few tens of nanometers for the typical AFMs, like orthoferrites [30], (40 nm for ω 0 /2π = 500 GHz, c = 20 km/s).The ex-arXiv:2403.13949v1 [cond-mat.mes-hall]20 Mar 2024 citation of such short coherent waves is a fundamental problem of modern magnonics since it requires an ultracompact source of magnons [31], despite different finesses, such as the usage of higher-order radial and azimuthal modes.
Here, we propose to employ a spin-current driven domain wall (DW) in an AFM as an ultra-compact source of the propagating coherent SWs.We demonstrate theoretically and by micro-magnetic simulations that the simple spin texture, such as an AFM DW, driven by spin current [32,33], can be a source of the propagating SWs with substantially high frequencies and short wavelengths, comparable to the exchange length of the AFM.We consider a device, schematically shown in Fig. 1, which is based on a thin film of an AFM with easyaxis anisotropy and n-fold rotational symmetry in the hard plane.In the proposed generator, the spin current flowing from the adjunct layer with the polarization along the principal axis excites the precession of the Néel vector within the DW.We assume the finite size of the spin current source with a width L located directly under a DW.The threshold current of the excitation is defined by the value of the anisotropy in the hard plane, and the frequency of the DW precession ω is tuneable by the strength of the spin current.We show that the above precession of the DW leads to the excitation of two modes of magnons with the frequencies (n ± 1)ω, where n is the order of the anisotropy.A robust emission of the propagating SWs into the AFM strip occurs when (n ± 1)ω > ω 0 , where ω 0 defines the frequency of AFM resonance.Consequently, the maximum achievable frequency of SWs is (n + 1)ω, which corresponds to very short wavelengths of the SW, comparable with the exchange length, especially for the hexagonal AFMs.The excitation of the high wavevectors is possible due to the substantially small width of the DW in AFM, which is hard to achieve by any other method.
Model.-The low-energy dynamics of a colinear AFM can be described using the Lagrangian L = T − U for the Néel vector l = (M 1 − M 2 )/M s , where |M i | = M s /2 is the magnetization of the sublattice i = 1, 2 and M s is the value of saturated AFM magnetization.The "kinetic" energy T = (M s /2γω ex )(∂ t l) 2 determines the inertial properties of the AFM spin dynamics, where ω ex = γH ex is the frequency defined by the exchange field H ex of the AFM, and γ is a gyromagnetic ratio.The "potential" term U = (A/2)(∇l) 2 + w a (l) is determined by nonuniform exchange (A) and anisotropy energy w a .Expressing the Néel vector in spherical coordinates l = {sin θ cos ϕ, sin θ sin ϕ, cos θ}, the anisotropy energy density reads: where the first term defines uniaxial anisotropy of the easy-axis type (K > 0), and the second one defines an anisotropy in the hard plane (K n > 0), for an AFM with an n-fold axis C n .Here z-axis is chosen along the easy axis of the AFM, K > K n , and the ground state corresponds to l z → ±1, (θ = 0, π).
A purely uniaxial AFM model (K n = 0) possesses formal Lorentz invariance [28,34] with the characteristic velocity c = γ H ex A/M s and degeneracy of the antiferromagnetic resonance (AFMR) frequency ω 0 = γ H ex K/M s .The solution for a stationary DW with boundary conditions l z | ±∞ → ±1 can be found from the minimum of the potential energy U as cos θ 0 = tanh x/x 0 and φ = φ s , where x 0 = A/K is the thickness of stationary DW and angle ϕ s determines the rotation of the l vector in the hard plane.The rotational dynamics of interest thus can be described by the transformation , where ω denotes angular velocity of the Néel vector precession in a DW [28,34].To induce the rotational dynamics, a spin current that is polarized along the easy axis of the AFM can be utilized.The frequency ω dependence on the current j is governed by the equilibrium between the total energy losses in the DW and the energy gained within the constrained region (with the width L, see Fig. 1) of the spin current's contact area [32]: where α is an effective Gilbert damping, σ is a spin torque efficiency and j is a density of electric current.In the case of a large spin-torque source, L ≫ x 0 and ω ≪ ω 0 , the frequency of the rotation is linearly proportional to the applied current ω = σj/α.Let us continue our analysis with the second term of Eq. 1 -namely, anisotropy in the hard plane K n , that leads to the excitation of spin waves.In order to investigate spin waves excitation, we consider small perturbations of the initial DW solution as θ = θ 0 (x) + ϑ(x, t) and φ = φ s + ωt + µ(x, t)/ sin θ 0 (x).It is convenient to combine polar and azimuthal perturbations into a single complex variable ψ = µ + iϑ.Assuming small value of the symmetry-reducing term K n ≪ K, one can obtain the linearized equation for ψ in the form [35] (see Supplementary Material [36] for the details): where Ĥ0 is the Schrödinger operator with the reflectionless Pöschl-Teller potential Ĥ0 = −∂ 2 ξ + 1 − 2 cosh 2 ξ, ξ = x/∆.The left side of Eq. ( 3) describes smallamplitude excitations in an AFM containing a precessing DW.The included type of potential created by a DW for linear SWs was already discussed for the FM as well as AFM materials.Particularly, the emission of exchange SWs from a Bloch DW, excited by a microwave magnetic field, was predicted for FMs in [37].For uniaxial AFMs, the above approach was employed in [38], where the propulsion of a DW by incoming SWs was demonstrated.The right-hand side of Eq. ( 3) represents periodic driving "force" with frequencies ±nωt and corresponding amplitudes B ± n (ξ) = −iB n sin n−1 θ 0 (cos θ 0 ± sin θ 0 )/2, where ) is proportional to the value of the anisotropy in the hard plane.Please note that superscripts indicate the sign of the corresponding frequency, i.e., the direction of ψ rotation.
The free solution of Eq. ( 3) can be represented in the form of a planar wave ψ = e i( kξ+ Ωt) , where k = k∆ is the rescaled wave vector and Ω = Ω±ω is the wave frequency in the rotating reference frame.At a large distance from the DW, k and Ω are connected by the relation which is a transformed version of the known dispersion law for spin waves Ω 2 = ω 2 0 + c 2 k 2 in the observer's coordinate system.
As it follows from the right-hand side of the Eq. ( 3), ψ(ξ, t) should be expressed as a linear combination of terms with both positive and negative frequencies ±nωt.However, it is sufficient to consider one frequency sign since the part with the opposite sign is symmetric with respect to ξ = 0. Separating spatial and time variables as ψ = χ ± (ξ)e ±inωt the equation ( 3) can be written as for the spatial part χ ± (ξ), where U ± ω (ξ) is a dimensionless potential for SWs created by a DW rotation and is given by The function B ± n (ξ) in Eq. 5 defines the amplitude of a spin wave, while the potential U ± ω (ξ) defines the condition for its propagation.Particularly, for the propagating SW in the form χ(ξ) ∝ e ±i kξ the wavevector acquires the real value k2 > 0 when Thus, Eq. ( 7) is a condition for a SW propagation with a wavevector k2 given by the relation (4) with a substitution Ω → nω.
The potential U ± ω depends on the frequency of DW rotation ω, see Fig. 2, which in turn can be controlled by the applied current in accordance with Eq. 2. Thus, by increasing the current, the condition for the emission is fulfilled when certain critical frequencies are exceeded: FIG. 2. Potential U ± ω created by a precessing DW for SWs, given by Eq. ( 6), for a different frequency ω of a DW precession.Solid lines correspond to the positive sign of a SW frequency, while dashed lines correspond to the negative sign.
Micromagnetic simulations.-Tovalidate our analytical findings, we carried out micro-magnetic simulations using MuMax3 solver [39] for a system schematically shown in Fig. 1.The AFM film has lateral sizes 0.14 × 1.56 µm 2 with a thickness of 5 nm.The selected parameters used for the AFM correspond to the DyFeO 3 and are given as [30]: α = 10 −3 , M s = 8.4 • 10 5 A/m, A = 18.9 pJ/m, H ex = 670 T, the anisotropy constant along the easy axis K = 300 kJ/m 3 .These parameters correspond to the characteristic speed c = 22 km/s, the frequency of the AFM resonance ω 0 /2π = 0.45 THz, and the width of a stationary DW x 0 = 8 nm.DyFeO 3 was chosen due to the relatively simple tunability of the second anisotropy K 2 in this material, for example, by temperature [40][41][42].Particularly, at low temperatures, it is possible to achieve a uniaxial state [40] with K n = 0, where two magnon modes are degenerated, and by varying temperature in the vicinity of this point, it is possible to tune K 2 in a wide range.
A spin current source with a lateral size 0.14 × 0.1 µm 2 (L = 100 nm) is positioned under a DW at the center of the device, injecting a spin torque polarized along the easy axis of the AFM.The frequency of the DW rotation is evaluated at the central location, where an initially relaxed DW is present.The selected polarization of a spin torque does not induce translational DW movement.
However, additional methods, such as nanoconstrictionbased pinning, can be utilized for the DW stabilization, if necessary.The frequencies of the excited SWs are measured at a distance of 300 nm from the DW. Figure 3 shows the results of simulations with 2-fold anisotropy in the hard AFM plane, considering various values of K 2 .The presence of K 2 anisotropy induces the excitation threshold current σj th = ω 2 2 /(2ω ex ), where ω n = γ H ex K n /M s .In particular, for K 2 /K = 0.02, the excitation starts at j th = 0.53 × 10 12 A/m 2 with frequency ω th ≃ σj th /α = ω 2 n /(2αω ex ) ≈ 100 GHz.With an increase in current, the frequency of DW rotation reaches the critical frequency ω cr,1 = ω 0 /3 = 150 GHz, leading to the detection of SWs at a large distance from the DW.The dependence of the DW frequency on the applied current is in good agreement with Eq. 2, despite that it is derived with the assumption K n /K ≪ 1.As predicted above, the frequency of the propagating SW is multiple of the DW frequency with a factor of n + 1 = 3 and, hence, follows the scaled dependence (2) on the applied current.
Since increasing the anisotropy K n leads to an increase in the threshold current j th , the frequency of a DW precession at the threshold exceeds ω cr,1 for high values of K n .As a result, only SWs with a substantial frequency gap above AFMR can be excited in this case, see K n /K = 0.04 and 0.06 in Fig. 3. Another outcome of raising K n is the increase of the SW amplitude, since driving term B ± n ∝ K n in Eq. ( 5).The SW radiation serves as an additional dissipation mechanism and results in a reduction of the measured DW frequency (and correspondingly the frequency of emitted SWs) as compared to the dependency (2).This effect is visible in Fig. 3 for The results for 4-fold anisotropy with K 4 /K = 0.02 are shown in Fig. 4. Here, all other parameters are left unchanged for the possibility of a direct comparison with the n = 2 case.The excitation threshold current for n = 4 is given by σj th = ω 2 4 /(3ω ex ), which corresponds to j th = 0.35×10 12 A/m 2 and a frequency of ω th /2π = 70 GHz.Upon reaching the first critical frequency ω cr,1 = ω 0 /5 = 90 GHz, only SWs with a five-fold frequency are observed.As the current is further increased, the DW surpasses the subsequent critical frequency ω cr,2 = ω 0 /3 = 150 GHz, leading to the emission of SWs with a triple frequency as well.
Fig. 4 b) and c) show group velocity and a wavelength of emitted SWs as a function of applied current.Our simulation results suggest ultra-high velocities, exceeding 10km/s, even at low supercriticality, while at higher currents, the velocity of emitted SWs is closely approaching the maximum value of 22km/s.Such a result is extremely hard to achieve by any other method of excitation due to the extremely small wavelength ≃ 10nm (see Fig. 4 c) of the magnon.
Discussion.-One can note that Eq. ( 3) is derived for the conservative case, i.e., does not take dissipation and spin current into account.Thus, the emission of the SWs can be created by any mechanism, which leads to the corresponding spin precession in the DW, and spin torque induced by a current is one of them.Gilbert damping defines the frequency of the DW precession in accordance to Eq. ( 2) and also leads to the decay of the propagating SWs, as one can see in the inset of Fig. 3.
It's worth mentioning that the anisotropy in the hard plane is not the only mechanism that leads to the reduction of the DW dynamic symmetry [35].The corresponding effect of the SWs emission can occur in AFM with a specific form of the Dzyaloshinskii-Moriya interaction (DMI) characterized by a function D(θ, ϕ).The forms of the functions D(θ, ϕ) for many AFMs are detailed in Ref. [43].The incorporation of DMI results leads to the term of a form D(θ) sin nωt in the right-hand side of Eq. ( 3), which acts as a periodic driving "force", similarly to the effect of anisotropy.
To summarize, it has been shown theoretically and confirmed by micromagnetic simulations that the AFM DW, in which internal rotational dynamics is excited by a spin current, can be utilized as a generator of SWs with remarkably high frequencies and group velocities, which correspond to short wavelengths of the order of the AFM exchange length.The AFM DW, due to its small characteristic width, serves as an efficient generator of SWs that are difficult to excite by other methods.In addition, the application of such radiation for the synchronization of AFM oscillators with multiple DWs, where the dynamics is induced by the spin torque, is of particular interest.This project is partly funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant TOPSPIN No 835068) and the Swedish Research Council Framework Grant Dnr.2016-05980.
Linearizing the left-hand sides of the system ( 12) and ( 13) with respect to ϑ and µ and limiting to the zeroth approximation in the right-hand sides, one obtains a system of linear inhomogeneous equations for ϑ and µ in the following form: Here, the terms that do not contain time dependencies are omitted, and the following notation is used ).The equations for ϑ and µ contain the Schrödinger operator with the Pöschl-Teller potential For further analysis, it is convenient to introduce a complex variable ψ = µ + iϑ.By using the transformation one gets the equation for ψ in the following form: For a free solution of the form ψ ∝ e i( kξ+ Ωt) , the Lorentz-transformed wave vector k = k∆ = kx 0 / 1 − ω 2 /ω 2 0 and the wave frequency in the rotating frame Ω are connected at a distance from the DW by the reletation From Eq. ( 18), the solution for ψ should include additives with positive and negative frequencies ±nωt.The general solution can be represented as a superposition of solutions with different frequency signs.It is sufficient to consider the part of the equation that is proportional to e inωt , since the solution that is proportional to e −inωt is symmetric with respect to ξ = 0.The equation for the coordinate part of the function ψ = χ(ξ)e +inωt reads To simplify, we replace the expressions of type sin n θ 0 by the delta function sin n θ 0 − → u n δ(ξ), and the expressions of type sin n−1 θ 0 cos θ 0 by the derivative of the delta function with respect to the coordinate sin n−1 θ 0 cos θ 0 − → u n−1 δ ′ (ξ)/(n − 1) , where the constant u n is determined from the condition ∞ −∞ sin n θ 0 dx = ∞ −∞ sech n ξdξ = u n , and we also approximate cos θ 0 by a step function Ξ, putting Ξ = sign ξ.Formally, these simplifications are only applicable for very large k >> 1; however, the numerical calculation showed that the results obtained in this approximation are qualitatively applicable to k ∼ 1.
The solution of the coordinate part thus can be found as a superposition of the solutions with separately considered delta function and a derivative of the delta function in the form: Le −ik (−) ξ at ξ < 0, Re ik (+) ξ at ξ > 0, (22)

FIG. 1 .
FIG. 1. Schematic representation of the proposed ultrashort SW generator.The spin torque source with a width L is positioned at the device's center beneath the AFM DW.The spin current polarization p aligns with the easy axis.The spin current application induces a precession of the Néel vector within the DW in the hard plane with n-fold symmetry.This precession results in the emission of SWs, as schematically shown in the lower-left corner.

FIG. 3 .
FIG. 3. The results of simulations for two-fold anisotropy in a hard AFM plane, n = 2. a) The frequency of the DW rotation (empty circles) and emitted SWs (filled circles) are shown as a function of the applied current density for different values of K2 anisotropy.Solid black lines are calculated analytically using Eq.(2).Angular frequency labels are employed for simplicity and correspond to the respective rotational frequencies f = ω/2π b) The profiles of emitted SWs far from a DW with the extracted values of the wavelengths: λ = 71 nm (red), 40 nm (green), 25 nm (blue).

FIG. 4 .
FIG. 4. The results of simulations for four-fold anisotropy in a hard AFM plane, n = 4. a) The frequency of the DW rotation (empty circles) and emitted SWs (filled circles) are shown as a function of the applied current density for K4/K = 2 × 10 −2 .Solid black lines are calculated analytically using Eq.(2).b) Group velocity and c) wavelength of the excited SWs as a function of the applied current density.