Terahertz Néel spin-orbit torques drive nonlinear magnon dynamics in antiferromagnetic Mn2Au

Antiferromagnets have large potential for ultrafast coherent switching of magnetic order with minimum heat dissipation. In materials such as Mn2Au and CuMnAs, electric rather than magnetic fields may control antiferromagnetic order by Néel spin-orbit torques (NSOTs). However, these torques have not yet been observed on ultrafast time scales. Here, we excite Mn2Au thin films with phase-locked single-cycle terahertz electromagnetic pulses and monitor the spin response with femtosecond magneto-optic probes. We observe signals whose symmetry, dynamics, terahertz-field scaling and dependence on sample structure are fully consistent with a uniform in-plane antiferromagnetic magnon driven by field-like terahertz NSOTs with a torkance of (150 ± 50) cm2 A−1 s−1. At incident terahertz electric fields above 500 kV cm−1, we find pronounced nonlinear dynamics with massive Néel-vector deflections by as much as 30°. Our data are in excellent agreement with a micromagnetic model. It indicates that fully coherent Néel-vector switching by 90° within 1 ps is within close reach.

)).The dotted black lines indicate the linear approximation of the model.(e) Odd-in- signal Δ(, + 0 ) for  pk = 15 kV/cm and  s =  pr = 0° (blue) and 90° (red).The inset indicates the probe polarization in the lab frame.For these measurements, a second Al2O3 substrate was used to compensate the birefringence of the first substrate.(f) Time-average of signal over gray area in panel (e) vs incident probe polarization angle  pr (blue circles) and reference curve cos(2 pr ).The inset indicates the probe polarization rotation in the lab frame.Signals can be found in Fig. S8a.16).The red dotted line shows the signal for  pk / 0 = 0.15 for comparison.(c) Calculated dynamics Δ  () for the three excitation amplitudes  (diamonds) in panel (a).Dots correspond to Δ  () at  = 0.6 ps (see panel (a)).For illustrative purposes, the Gilbert damping parameter is chosen smaller than the experimental value.(d) Simulated dynamics of the deflection angle Δ  for various driving fields  pk / 0 .Solid lines correspond to the fits from panel (b) ( pk / 0 ≤ 1), whereas dash-dotted lines are simulations for moderately larger fields ( pk / 0 > 1).Introduction.Antiferromagnets offer great potential for robust, ultrafast and space-and energy-efficient spintronic functionalities [1][2][3][4][5][6].Remarkably, in a recently discovered class of metallic antiferromagnets with locally broken inversion symmetry, coherent rotation of the Néel vector  should be possible by simple application of electrical currents.This fascinating phenomenon is driven by the Néel spin-orbit torques (NSOTs) [7][8][9] that arise from staggered spin-orbit fields at the antiferromagnetically coupled spin sublattices [10,11].So far, switching studies of CuMnAs and Mn2Au showed indications of NSOTs [12][13][14][15], but also a strong and possibly dominant heat-driven reorientation of  [16,17].
To reveal direct signatures of NSOTs and gauge their potential for ultrafast coherent antiferromagnetic switching, electric fields at terahertz frequencies are particularly interesting because they are often resonant with long-wavelength antiferromagnetic magnons [18][19][20][21][22][23].Consequently, terahertz NSOTs should require significantly smaller current amplitudes to modify antiferromagnetic order and, consequently, mitigate unwanted effects such as Joule heating.Studying terahertz NSOTs should also provide fundamental insights into key parameters such as torkance and the frequency and lifetime of long-wavelength magnons in the novel antiferromagnets.
Probing antiferromagnetic responses is highly nontrivial because, unlike the magnetization of ferromagnets, the Néel vector  cannot be controlled by external magnetic fields below several tesla in most compounds [24][25][26].Consequently, experimental separation of magnetic and nonmagnetic dynamical effects is challenging [27,28].Moreover, the small size of antiferromagnetic domains [24,29] implies that the spatial average of the Néel vector and the NSOTs vanishes.
In this work, we report on terahertz-pump magneto-optic-probe experiments on Mn2Au thin films (Fig. 1a).We find birefringence signals linear in the incident terahertz electric-field transient.They can consistently be assigned to a uniform, strongly damped and coherent antiferromagnetic magnon at 0.6 THz that is excited by field-like NSOTs.When the terahertz field inside the Mn2Au film exceeds 30 kV/cm, the Néel-vector dynamics become significantly nonlinear.Comparison to an analytical model allows us to extract values of NSOT torkance, magnon frequency and Gilbert damping, all of which are consistent with previous predictions and experiments.We deduce that the Néel vector  is transiently deflected by as much as 30° at the maximum peak field of 40 kV/cm.
Our results imply that terahertz electric fields and NSOTs can drive coherent nonlinear magnon dynamics in Mn2Au, thereby bringing coherent switching, a central goal of antiferromagnetic spintronics, in close reach.Indeed, extrapolation of our data indicates that coherent rotation of  by 90° can be achieved by terahertz pulses with a moderately increased peak field strength of around 120 kV/cm inside Mn2Au.
Experiment.We excite Mn2Au thin films with intense phase-locked single-cycle terahertz pulses to drive spin dynamics and magneto-optically monitor them with a femtosecond probe pulse (Fig. 1a).
We study two epitaxially grown Mn2Au(001) thin films (thickness of 50 nm) with an Al2O3 cap layer (3 nm) on Al2O3(11 ̅ 02) substrates (500 mm).In the as-grown samples, the four equilibrium directions of the Néel vector  0 are at 0°, 90°, 180° and 270° relative to  1 [24,29].The resulting domains have a size of the order of 1 mm with approximately equal distribution over the film plane and a small strain-induced preference along one easy axis [33].Following growth, one sample was subject to an intense magneticfield pulse [24,34,35], which aligned most domains at 0° or 180° (Fig. 1c).For test purposes, we also consider Mn2Au|Py samples in which the volume-averaged Neel vector 〈 0 〉 is oriented parallel to the magnetization of the exchange-coupled ferromagnetic Py (permalloy Ni80Fe20) layer [29].
In our ultrafast setup (Figs.1a, S1, S2), the terahertz pump and optical probe beam are both normally incident onto the sample.The driving terahertz field  = (, ) contains the electric component  =   , shown in Fig. 2a, and the magnetic component  =   , which exhibits the same shape.In free space,  reaches peak values up to 600 kV/cm, which is reduced to 6% inside the Mn2Au film.The resulting spin dynamics are monitored by linearly polarized optical probe pulses [36] with the polarization plane at an angle  pr relative to the laboratory axis   (Fig. 1a).The detected signals Δ(, ,  0 ) are the pump-induced probe-polarization rotation and ellipticity vs pump-probe delay .The sign of the pump field (±) and the local Néel vector (± 0 ) can be reversed by, respectively, a pair of polarizers and rotation of the sample azimuth  s = ∢( 1 ,   ) by 180°.
When the terahertz field is reversed, the induced signal Δ(, −,  0 ) (blue curve in Fig. 2a) changes sign, too, as expected for NSOTs.Therefore, we consider the signal component that is odd with respect to , i.e., Δ(,  0 ) = Δ(, +,  0 ) − Δ(, −,  0 ) 2 . (1) Fig. 2b displays Δ(, ± 0 ) for opposite local Néel vectors.Again, the two signals are approximately reversed versions of each other, pointing to a strong contribution of the antiferromagnetic order.Analogous to Eq. ( 1), we determine signals Δ() odd in both the pump field  and the Néel vector  0 and focus on them in the following.
To check further that Δ() reports on magnetic order, Fig. 2c displays waveforms Δ() from the prealigned and as-grown Mn2Au film.Remarkably, the signal from the prealigned sample is more than a factor of 5 larger and independent of the probed spot over areas much larger than the individual antiferromagnetic domains.In contrast, Δ() from the as-grown sample is typically within the experimental noise floor (see Fig. S5) and only exceptionally large for the example waveform depicted in Fig. 2c.For the bare Al2O3 substrate, the signal odd in  is zero within the experimental accuracy.
The different response of the as-grown and prealigned sample (Fig. 2c) and the very similar response of the prealigned and Mn2Au|Py test sample (Supplementary Note 11) provide strong evidence that the component Δ() is related to the antiferromagnetic order of Mn2Au.
Fig. 2d shows that the absolute maximum of Δ() grows roughly linearly with the peak amplitude  pk of the terahertz electric field inside the sample over the full range of  pk .In contrast, the absolute minimum of Δ features an onset of nonlinear behavior for  pk > 30 kV/cm (see black-dotted lines and Fig. S3).Therefore, the signals in Fig. 2b are the linear response to the driving terahertz field (|| < 15 kV/cm), and we will first focus on this lowest-order perturbation regime.
Signal phenomenology.To understand the pump-probe signal, we need to relate it to the perturbing terahertz pump field  = (, ) and the instantaneous magnetic order, which is quantified by  and .
The symmetry properties of Mn2Au impose strict conditions on the relationship between Δ and , , , as detailed in the Methods.In particular, we find that any signal linear in  and odd in  0 (Fig. 2c) arises entirely from the terahertz electric field  and not the magnetic field .
Generally, the signal depends on pump-induced changes in , , and the non-spin degrees of freedom.Our experimental geometry (Fig. 1a) and the point-symmetry group of Mn2Au [31] imply that the pumpinduced signal is up to second order in  and  given by the form Here, ,  and  are sample-and setup-dependent coefficients,  pr −  s = ∢( pr ,  1 ) (Fig. 1a), and  ∥ =  − (  ⋅ )  is the in-plane projection of the Néel vector with azimuthal angle   = ∢( ∥ ,  1 ).A preceding Δ denotes pump-induced changes, and 〈. 〉 means spatial averaging over the probed Mn2Au volume.The first and second term of Eq. (2) are quadratic in  ∥ (magnetic linear birefringence (MLB) [28,33]) and monitor spin dynamics through changes in  ∥ 2 and   .The third term is linear in  (magnetic circular birefringence (MCB) [28]) and reports on out-of-plane variations of .
Note that the last term Δ  is unrelated to transient changes in  and .It exclusively arises from variations of the non-spin degrees of freedom , such as phonons, and can be shown to exhibit the same dependence on   ,   0 and  s as the first three terms (see Methods).Nonmagnetic contributions of this kind are rarely considered [37].Figs.S4 and S10, however, show that Δ  does not make a dominant contribution to our total signal Δ().
To identify the dominant terms in Eq. (2), we vary the incoming probe polarization  pr while keeping  s = 0° (Fig. 2e).We find that the waveforms Δ(,  0 ) exhibit opposite sign for  pr = 0° and 90° (Fig. 2e).Their amplitude follows a cos(2 pr ) dependence to very good approximation (Fig. 2f).Therefore, the signal predominantly derives from the second term of Eq. (2).Assuming the pump-induced changes in  ∥ 2 and   are small, we linearize this term.In the prealigned sample, we have si (2  0 ) = 0 for both the 0° and 180° domains, and Eq.(2) simplifies to This important result implies that the signal Δ directly monitors the dynamics of the spatially averaged azimuthal rotation Δ  of , with the rotation angle being odd in  0 and odd in .
Note that a nonzero 〈Δ  ()〉 requires a non-vanishing average Neel vector 〈 0 〉 in the probed volume.
We can understand the occurrence of 〈 0 〉 ≠ 0 by the magnetic prealignment procedure (see Methods and Supplementary Note 5).

NSOT-driven terahertz magnon.
As the signal Δ() cannot arise from the terahertz magnetic field, we can discard Zeeman torque (∝ ) and field-derivative torque (∝ /) [38,39] as possible microscopic mechanisms.Effects of Joule heating are excluded, too, as they would scale quadratically with the pump field.
Because the Néel-vector rotation Δ  is linear in the terahertz electric field , it may arise from bulk NSOTs.To put this conjecture to the test, we consider the effect of NSOTs on the magnetic order of Mn2Au.As detailed in Fig. 3a for a single  0 domain, NSOTs induce an in-plane deflection Δ  ∝   ⋅ ( 0 × Δ) of the Néel vector.As the NSOT-induced change Δ is even in  0 (Fig. 3a), Δ  is odd in  0 , precisely as Δ() (Eq.( 3)).The amplitude of Δ  is proportional to cos ∢(,  0 ).To test whether the measured Δ() follows the same dependence, we rotate the sample and, thus, the Néel vector (Fig. 1a).We find that Δ() indeed scales with cos ∢(,  0 ), as shown in Fig. 3b, c.
To summarize, the signal Δ() ∝ 〈Δ  ()〉 is fully consistent with the terahertz-NSOTs scenario of Fig. 3a owing to its phenomenology: linear in , odd in  0 and scaling with cos ∢(,  0 ).The resulting predominant in-plane motion of  =  0 + Δ corresponds to the in-plane magnon mode of Mn2Au.It is accompanied by an out-of-plane magnetization |Δ| ∝ |Δ/| (Fig. 3a) [40,41], which is more than 2 orders of magnitude smaller than Δ and below our detection sensitivity.The remaining second magnon mode of Mn2Au [32,40], which involves an out-of-plane oscillation of , would be even in  0 and possibly be masked by contributions independent of magnetic order (see Methods).
Micromagnetic model.From Fig. 3a, we expect a harmonic time-dependence of Δ  that starts sinelike.Indeed, as detailed in the Methods, the rotation Δ  can be described as deflection of a damped oscillator [40,42] whose potential energy  ani ∝ − ani  ex cos(4  ) (Fig. 4a) is determined by the inplane anisotropy field  ani and exchange field  ex of the Mn2Au thin film.The damping of the oscillator is proportional to  ex and the Gilbert parameter  G .The driving force scales with  NSOT (), where  NSOT is the NSOT coupling strength (or torkance), and  ≈ 1.5 S/m is the measured terahertz conductivity (Fig. S6).
As a cross-check, we solve Eq. ( 4) for () by numerical deconvolution without model assumptions.We find that the deconvoluted response agrees well with the fit-based result in both the time (Fig. 3e) and frequency domain (Fig. 3f).The amplitude spectrum of the transient deflection Δ  illustrates the broad resonance-like response given by  (Fig. 3f).
Our experimentally obtained magnon frequency lies outside the accessible range of previous Brillouinand Raman-scattering measurements [32].A peak at 0.12 THz was ascribed to the in-plane magnon mode, but the magnetic origin of this feature is not confirmed.The bare magnon frequency  0 /2 allows us to estimate the spin-flop field by  sf ∼  0 /, where  is the gyromagnetic ratio (see Methods).We infer  sf ∼ 20 T at a temperature of 300 K, which is consistent with a previous order-of-magnitude estimate of 30 T at 4 K [24,43].A more accurate comparison requires a detailed understanding of the magnetic reordering processes at high magnetic fields, which remain elusive.With  ex = 1300 T [31], we extract a Gilbert-damping parameter of  G = / ex = 0.008, which is consistent with theoretical predictions for metallic antiferromagnets [44,45] and recent studies of optically driven spin dynamics in IrMn [46].
Nonlinear regime and torkance.Finally, we increase the peak terahertz field  pk inside the sample above 30 kV/cm to study the nonlinear response of Mn2Au indicated by Fig. 2d.Examples of signal waveforms Δ(), normalized to  pk , are shown in Fig. 4b.Remarkably, as  pk grows (red to orange curve), the normalized signal not only decreases its peak value but also becomes more symmetric.
Qualitatively, this transition from a linear to nonlinear response can be well understood by the anharmonic potential of Fig. 4a.As  ani grows sub-quadratically for large deflection angles Δ  , smaller restoring forces and, thus, slower dynamics than in the harmonic case result.Importantly, in the anharmonic regime, the temporal waveform is unambiguously connected to the excitation strength, as illustrated for the fictitious impulsive driving forces in Fig. 4c.Further, as the magneto-optic signal is governed by the second term of Eq. (2), it exhibits a sub-linear growth at large deflection angles Δ  .
Quantitatively, we fit the measured signals Δ() ∝ 〈si [2Δ  ()]〉 (Fig. 4b) by numerically solving the micromagnetic model (see Methods), where a scaling factor and the torkance  NSOT (Eq.( 16)) are the only free fit parameters.As the dynamics of the 0° and 180° domains are given by ±Δ  (), the spatial average Δ() reflects the dynamics of a single domain, but just with decreased signal amplitude.The inferred  NSOT = (150 ± 50) cm 2 A s ⁄ is, to our knowledge, a first experimental determination of an NSOT torkance.It corresponds to a staggered field of (8 ± 3) mT per 10 7 A cm 2 ⁄ driving current density, which agrees well with ab initio calculations that found 2 mT per 10 7 A cm 2 ⁄ [11].Our procedure also allows us to calculate the signal amplitudes vs the terahertz peak field  pk .As seen in Fig. 2d, the onset of non-linearity is in good agreement with our experiment.The calculated Néel-vector dynamics Δ  () of a single domain are shown in Fig. 4d.We find that the rotation angle Δ  reaches 30° at the maximum terahertz peak field of 40 kV/cm inside the sample.This massive deflection is about 2 orders of magnitude larger than the magnetization deflection in ferromagnets that were induced by incident terahertz pulses comparable to ours [47,48].It illustrates the benefits of NSOTs and exchange-enhancement of the resonant Néel vector response in antiferromagnets.
Our modeling enables us to extrapolate the dynamics to even higher driving fields (Fig. 4d).Remarkably, at a peak amplitude exceeding our maximum available incident field of 600 kV/cm by only a factor of 2.5, the Néel vector overcomes the potential-energy maximum of the magnetic anisotropy at   = 45° (Fig. 4a) and coherently switches from   = 0° to 90°.The ultrafast switching time of only 1 ps is given by half the period   ⁄ of the terahertz magnon.We estimate that the temperature increase due to terahertz pulse absorption is less than 5 K (Supplementary Note 9).Therefore, resonant NSOTs induced by terahertz electric pulses are an ideal driver to achieve coherent ultrafast and energy-efficient antiferromagnetic switching.When the terahertz field is increased by a factor of 3.0, we even obtain switching to   = 180° (Fig. 4d).

Conclusion.
It is shown that terahertz electric fields exert field-like NSOTs in Mn2Au antiferromagnetic thin films.Using magnetic linear birefringence, we observe a strongly damped precession of the Néel vector in the sample plane at 0.6 THz.Our interpretation is consistent with regard to the symmetry and dynamics of our signals as well as their dependence on the terahertz field amplitude and magneticdomain structure of our samples.In particular, the torkance inferred by comparison with a spin-dynamics model agrees well with ab initio predictions.The maximum deflection of the Néel vector  currently amounts to as much as 30°, showing that coherent ultrafast switching of  by 90° without the need for heating is within reach.
Our study has profound implications for future antiferromagnetic memory applications at high speeds and minimized energy consumption and might even serve as a blueprint for similar functionalities in multiferroic materials that inherently feature a linear coupling between electric and magnetic order [49].Finally, the magneto-optic signals are significantly stronger in single-domain films (Supplementary Note 11), making them interesting candidates for spintronic detection of terahertz electromagnetic pulses.

Fig. 1 :
Fig. 1: Schematic of experiment and samples.(a) A linearly polarized phase-locked terahertz pump pulse (red) with electric field  =   is normally incident onto an antiferromagnetic Mn2Au thin film.The resulting spin dynamics are monitored by an optical probe pulse (blue) with field  pr by measuring its polarization change Δ (rotation and ellipticity) vs delay time  behind the sample.The probepolarization angle  pr = ∢( pr ,   ), sample azimuth  s = ∢( 1 ,   ) and terahertz field polarity (±) can be varied in the laboratory frame (  ,   ,   ).The sample-fixed frame is given by  1 = [110]/√2,  2 = [1 ̅ 10]/√2 and  3 =   = [001].(b) Schematic of the Mn2Au unit cell seen along   =  3 .The local magnetic moments on the Mn A and Mn B sites lead to the sublattice magnetizations  A and  B , respectively.The Néel vector  =  A −  B (purple arrow) equals  0 before pump excitation.For simplicity, the Au atoms are omitted.(c) In the magnetically prealigned sample, the in-plane distribution  0 (, ) consists of regions with  0 ↑↑ + 1 and  0 ↑↑ − 1 , resulting in a pattern of 180° domains.Dashed arrows indicate a possible pump-induced deflection Δ.(d) An in-plane electric field  induces a staggered spin-orbit fields  A SO = − B SO .For  ∥  0 , the resulting NSOTs on Mn A and Mn B moments are maximum and directed out-of-plane (∥  3 ). Fig.2

Fig. 2 :
Fig. 2: Signatures of terahertz spin dynamics.(a) Pump-induced probe polarization rotation Δ(, +,  0 ) (red) and Δ(, −,  0 ) (blue) vs pump-probe delay  measured for opposite polarities of the terahertz pump field  = (, ) in the prealigned sample with  s =  pr = 0°.The grey-shaded area shows the terahertz electric field () for reference.The incident peak terahertz field in air is 250 kV/cm, corresponding to  pk = 15 kV/cm inside the sample.(b) Signal components Δ(, + 0 ) ( s = 0°, orange solid line) and Δ(, − 0 ) ( s = 180°, blue) odd in the driving terahertz field (Eq.(1)).(c) Signal component Δ() = [Δ(, + 0 ) − Δ(, − 0 )]/2 odd in both driving field  and Néel vector  0 for prealigned (blue line) and as-grown (red) sample.The black line shows the signal from the bare Al2O3 substrate.(d) Maximum (red dots) and minimum value (blue dots) of signal Δ() odd in  and  0 vs peak terahertz electric field  pk inside the sample.Data was taken from a different sample region than in panels (a)-(c), yielding smaller signal magnitudes.The red and blue solid line is a model calculation (Fig. 4 and Eq.(16)).The dotted black lines indicate the linear approximation of the model.(e) Odd-in- signal Δ(, + 0 ) for  pk = 15 kV/cm and  s =  pr = 0° (blue) and 90° (red).The inset indicates the probe polarization in the lab frame.For these measurements, a second Al2O3 substrate was used to compensate the birefringence of the first substrate.(f) Time-average of signal over gray area in panel (e) vs incident probe polarization angle  pr (blue circles) and reference curve cos(2 pr ).The inset indicates the probe polarization rotation in the lab frame.Signals can be found in Fig.S8a. Fig.3

Fig. 4 :
Fig.4: Nonlinear Néel-vector precession and extrapolation to switching.(a) Magnetic anisotropy energy  ani ∝ − ani  ex cos(4  ) vs azimuthal rotation   of the Néel vector (blue solid line) and its harmonic approximation (red dotted line).The red, blue and orange dot indicates, respectively, the position and anisotropy energy at  = 0.6 ps following fictitious impulsive excitation by () ∝ () with relative strength  = 0.2, 0.4 and 0.8.(b) Normalized measured signals Δ()/ pk for normalized terahertz peak fields  pk / 0 , where  0 = 40 kV/cm is the maximum available field inside the sample.Signals are vertically offset for clarity.The gray-shaded area shows the incident driving field () for reference.Black lines are a joint fit using the model of Eq. (16).The red dotted line shows the signal for  pk / 0 = 0.15 for comparison.(c) Calculated dynamics Δ  () for the three excitation amplitudes  (diamonds) in panel (a).Dots correspond to Δ  () at  = 0.6 ps (see panel (a)).For illustrative purposes, the Gilbert damping parameter is chosen smaller than the experimental value.(d) Simulated dynamics of the deflection angle Δ  for various driving fields  pk / 0 .Solid lines correspond to the fits from panel (b) ( pk / 0 ≤ 1), whereas dash-dotted lines are simulations for moderately larger fields ( pk / 0 > 1). Fig.4