Coherent THz spin dynamics in antiferromagnets beyond the approximation of the Néel vector Open Access

Exploring efficient routes for excitation of coherent spin waves with the shortest possible wavelength and the highest achievable frequency is one of the major challenges of today’s spintronics, magnonics, and magnetic data storage.^1–5 Among magnetic materials, antiferromagnets have the highest frequencies of spin dynamics in the THz range, which is a significant advantage for becoming a crucial ingredient of ultra-high speed spintronic devices.^6–11 It has been demonstrated that ultrashort laser pulses can be employed for the generation of coherent spin waves in practically all classes of magnetically ordered materials.^12–15 The wavelength of the optically excited spin wave is, in principle, defined by the size of the illuminated volume and can be further decreased by introducing inhomogeneities.^16,17 However, the spin waves thus generated will still have frequencies and wavevectors close to the center of the Brillouin zone. An appealing alternative approach for the generation of spin waves with larger k-vectors is based on concomitant excitation of the two counter-propagating waves.^5,18 In antiferromagnets, such mutually coupled pairs of spin waves excited over the whole Brillouin zone form a so-called two-magnon mode, which couples to light via Raman scattering,¹⁹ and can be triggered by ultrashort laser pulses.^20,21

Until now, the approximation, which neglects individual strongly coupled quantum mechanical spins and treats a magnet as a continuous medium, has been fundamental to understanding spin dynamics, including that at ultrafast timescales.²² Using such an approximation, the simplest two-sublattice antiferromagnet can be modeled as two antiferromagnetically coupled ferromagnets with oppositely directed magnetizations |M₁| = |M₂| = M₀, the dimensionless antiferromagnetic vector L = (M₁ − M₂)/2M₀, and the net magnetization M = (M₁ + M₂)/2M₀. The propagation of an antiferromagnetic spin wave with the wavevector k and the angular frequency Ω_k in this approximation is described as spatiotemporal variations of the orientations and lengths of L and M. It is clear that upon decreasing the wavelength and approaching the edge of the Brillouin zone, the approximation based on such macroscopic vectors should eventually fail since it is substantiated only if the characteristic length scale of spin wave is much larger than the interspin distance. Despite this fact, the breakdown has not been examined for experimentally observed laser-induced spin dynamics of antiferromagnets.

Here, using the example of the Heisenberg antiferromagnet RbMnF₃, we demonstrate that the spin dynamics corresponding to mutually coherent spin waves with large opposite wavevectors cannot be understood if the light–spin interaction is modeled in conventional terms of macroscopic magnetization M and antiferromagnetic vector L. Instead, we propose to model such spin dynamics using spin correlations. We derive the equation of motion for the spin correlation function and demonstrate that, unlike M and L, the spin correlations in antiferromagnets do not exhibit inertia. It means that there is an optimal pulse duration for excitation of the spin correlation dynamics, while the macrospin dynamics in the antiferromagnet can be excited impulsively even by an infinitesimally short pulse.

The excitation of the two-magnon mode relies on the perturbation of exchange coupling, and the symmetry of the latter is intrinsically linked to the crystal structure, while for the models in terms of M and L, an orientation of magnetic moments is important. Therefore, it was crucial to choose a material in which the orientation of the magnetic moments and the resulting antiferromagnetic vector L are distinctly different from the main crystallographic axes. Here, we select the cubic fluoroperovskite insulator RbMnF₃ (point group $m 3 ̄ m$⁠), which is very close to the isotropic 3D Heisenberg antiferromagnet below the Néel temperature T_N = 83 K.^23,24 The spins (S = 5/2) of the Mn²⁺ ions form two equivalent magnetic sublattices coupled antiferromagnetically. Very weak magnetic anisotropy aligns the antiferromagnetic vector L along one of the ⟨111⟩ directions,²⁵ as shown in Fig. 1(b). The two-magnon mode in RbMnF₃ was observed in spontaneous Raman scattering experiments [Fig. 1(a)] and has the E_g symmetry and frequency 4 THz at low temperature.^26–29 

To excite coherent spin dynamics corresponding to the two-magnon mode, we perform a femtosecond two-color magneto-optical pump–probe experiment in the transmission geometry, as shown in Fig. 1(c). The laser pulses with duration 45 fs and central photon energy 1.55 eV were emitted by the Ti:Sapphire regenerative amplifier at a repetition rate of 1 kHz and were split into pump and probe (ℏω_pr = 1.55 eV) beams with an energy per pulse ratio of 10:1. The pump pulses were tuned by using an optical parametric amplifier (OPA) to a central photon energy ℏω_p = 1.03 eV and a fluence of 1.2 mJ/cm² and then were focused normally onto the sample surface into a spot with a diameter of 500 μm. The pump and probe pulses were linearly polarized at angles α and β with respect to the x‖[100] axis, respectively. The angles were controlled by half-wave plates. The 620 μm-thick slab sample with the optical quality of the surfaces was cut perpendicular to the cubic z‖[001] axis from a single crystal of RbMnF₃ grown using the Czochralski method. The projection of the antiferromagnetic vector L on the xy sample plane makes an angle of 45° with the x axis, as shown in Fig. 1(c). The sample was placed in the continuous-flow liquid helium cryostat. The spin dynamics triggered by an intense pump pulse induces in the sample a certain type of optical anisotropy.¹⁴ The latter is detected using a balance detection scheme to measure the induced ellipticity of the polarization of the probe pulse as a function of the time delay Δt between pump and probe pulses. Note that in this detection scheme, only modulation of the medium parameters that is nearly homogeneous within the probed area is detected.

Figure 2(a) shows a temporal evolution of the pump-induced ellipticity in the antiferromagnetic phase of RbMnF₃. The pump and probe pulses polarizations were α = 0° and β = 45°. One clearly distinguishes oscillations damped within a few ps. The Fourier spectra of the time trace for T = 5 K reveal a clear resonance at 4 THz. The frequency and amplitude of these oscillations decrease upon heating up to T_N [Figs. 2(b) and 2(c)]. The frequency and its temperature behavior are in fair agreement with those of the two-magnon mode obtained in spontaneous Raman scattering experiments.^26–29 The detected two-magnon mode corresponds to the simultaneous excitation of pairs of mutually coherent spin waves with opposite wavevectors throughout the Brillouin zone dominated by the waves with the shortest wavelengths and the highest frequencies Ω_π/a, as shown in Fig. 1(a).^{19,26,30–32} Thus, we can confidently conclude that the experimentally detected oscillations correspond to the spin dynamics originating from the pairs of spin waves with opposite wavevectors at the edges of the Brillouin zone.

To comprehend the observed spin dynamics, we performed measurements of the pump-induced ellipticity at various angles of the pump α and probe β polarizations at T = 5 K (Fig. 3). The most pronounced oscillations of the probe ellipticity are observed when the pump polarization is along the crystallographic x and y axes (α = 0°, 90°) and the probe polarization is at an angle β = 45°, 135°. Moreover, it is clearly seen that the phase of the oscillations is shifted by π when either pump or probe polarization is rotated by 90° [Figs. 3(a) and 3(b)]. To gain further insights, we performed two series of experiments for fixed pump (α = 0°) and rotated probe (β = 0°…360°) polarization and for fixed probe (β = 45°) and rotated pump (α = 0°…360°) polarizations. The variations of the signed Fourier amplitudes of the obtained waveforms are fairly well described by cos 2α and sin 2β [Fig. 3(d)].

In contrast to the earlier reported results of ultrafast spin dynamics,²² the observed polarization dependencies cannot be explained in terms of the macroscopic magnetization M and antiferromagnetic vector L. In particular, it was argued that excitation of pairs of mutually coherent spin waves with opposite wavevectors in isostructural antiferromagnetic fluoroperovskite KNiF₃³³ modulates the length of the antiferromagnetic vector L at the two-magnon frequency homogeneously across the excitation area. The latter is detected in experiments as oscillations of magnetic linear birefringence at the same frequency. In KNiF₃, L is along one of the main crystallographic axes, e.g., the x axis, while in the case of RbMnF₃, the projection of L on the xy sample plane is rotated over 45° from the x and y axes. Nevertheless, the pump polarization dependences are exactly the same in both antiferromagnets and possess maxima when the pump is polarized along one of the main crystallographic axes. Moreover, as the maximum signal in RbMnF₃ is observed when the probe is initially polarized along or perpendicular to the projection of L on the sample plane, the corresponding optical anisotropy does not originate from the oscillations of the length of L and cannot be understood in terms of magnetic linear birefringence. The probing geometry could agree with the transverse dynamics of L detected via such effect, which, however, yields the paradoxical conclusion that L homogeneously oscillates at the frequency twice as high as the maximal spin wave frequency (supplementary material).

Solving this equation of motion for the impulsive perturbation $Δ H ̂$ gives precession of $C ̂ k$ with dominating frequency $≈2zJS/ℏ$ (z = 6 is the number of nearest neighbors), which corresponds to the frequency of the two-magnon mode [see Fig. 3(c) and Eqs. (19) and (21) in the supplementary material]. Thus, we derived the equation of motion for spin correlations, which describes the dynamics of this parameter, in agreement with our and previously reported experiments. We note, that, unlike M and L in antiferromagnets,³⁶ the dynamics of spin correlation is described by the differential equation of the first order, and hence, the spin correlations do not show inertia.

From Eq. (4), we see that the polarization dependence of the pump-induced spin correlations is dictated by the light-induced perturbations of the effective fields $−∂ H ̂ k /∂ C ̂ k$⁠. In the supplementary material, we demonstrate that for the experimental geometry considered, one then obtains a polarization dependence ∝cos 2α, in agreement with what is observed experimentally [Fig. 3(d)]. Phenomenologically, this polarization dependence can be understood as follows. From Eq. (1), one sees that an inequality E^ν ≠ 0 gives $∂Δ H ̂ /∂ C ̂ ( δ ν ) ≠0$⁠. It means that pumping the antiferromagnet by a laser pulse polarized along one of the main crystallographic axes brings the system out of the thermodynamic equilibrium, and finding the new equilibrium will result in a change of the spin correlations. Such changes in the spin correlation functions contribute to the dielectric permittivity as $Δ ε μ ν = ∂ 2 ⟨ Δ H ̂ ⟩ /∂ E ν ∂ E μ$⁠, which may lead to a pump-induced anisotropy ɛ^xx − ɛ^yy ≠ 0 in the initially optically isotropic cubic system. For the $m 3 ̄ m$ point group, we obtain the pump-induced change of the dielectric permittivity ɛ^xx − ɛ^yy = 2gE² cos 2α, where g = ξ^xxxx − ξ^yyxx and ξ is the phenomenological polar tensor.³⁷ In the range of low absorption, such a pump-induced change of the dielectric permittivity results in the ellipticity of the probe pulse, which obeys sin 2β dependence on the incoming probe polarization, observed in the experiment [Fig. 3(d)]. Hence, the model based on spin correlation reproduces the experimental observation in terms of symmetry of excitation and detection.

Furthermore, the model correctly describes the linear dependence of the effect on the pump pulse fluence [Fig. 2(d)] since the perturbation of the effective field in Eq. (4) is linear in the pump pulse fluence $∝ ( E ν ) 2$⁠. This contrasts with the result one obtains considering dynamics of L. Indeed, the spin wave frequencies Ω_k are much lower than those of the pump photons ω_p, and the spin wave excitation involves a nonlinear process of down conversion ω₁ − ω₂ = 2Ω_k [see Fig. 1(a)]. Hence, in the lowest order, a light-induced torque acting on M or L and the amplitude of the excited spin wave are linear with respect to the pump fluence, as observed for the optically driven spin waves with low k.^12,36 As the spin wave wavevectors k ≈ π/a are much larger than the wavevector of light q ≈ 0, they can be detected only in pairs such that k − k = q ≈ 0. Therefore, the signal detected by the probe pulse can only originate from a product of the spin-wave amplitudes. Hence, the detected signal must be quadratic with respect to the pump fluence (see the supplementary material).

In our model [Eqs. (1) and (4)], when the electric field of the pump pulse is along one of the main crystallographic axes, it off-resonantly excites virtual charge transfer transitions between Mn²⁺ and F¹⁻ ions and, thus, effectively changes the exchange interaction between Mn²⁺ spins.³⁸ As a result, the spin correlation function $⟨ C ̂ ( δ ) ⟩$ is impulsively altered along the corresponding crystallographic axis. In the following, the spin correlation function along this axis will oscillate at the frequency of two-magnon mode, which is defined by the energy of the exchange interaction taking twice. The correlations along the two other axes are affected as well since different correlations can comprise the same spins. Obeying the energy conservation law, spin correlations along the two other crystallographic axes oscillate with the opposite phase, and the total amplitude along all three axes equals zero. Naturally, such kind of dynamics cannot be comprehended in terms of macrospins L and M, which do not distinguish anisotropic changes of spin–spin coupling (supplementary material). Despite this fact, the dynamics manifests itself in the modulation of the symmetric part of the dielectric permittivity, which distinguishes the spin correlations along different axes. Thus, the spin dynamics is probed by measuring dynamic linear birefringence (for details, see the supplementary material). We stress that the considered scenario for excitation and detection of the two-magnon mode holds for the isostructural KNiF₃^21,33 as well. The model suggested in those papers remains valid but represents the next-order effect involving further lowering of the symmetry or strong magnetic anisotropy.³⁴ 

In conclusion, we have demonstrated a regime of ultrafast spin dynamics in antiferromagnetic RbMnF₃, where the models considering the antiferromagnetic vector L and the net magnetization M fail. Instead, we propose to describe the regime in terms of the spin correlation function, derive the corresponding equations of motion, and reveal that, unlike the macrospins, the spin correlation function in antiferromagnets does not possess inertia. As a consequence, a pump pulse with a duration equal to about 1/8th period of the spin wave at the edge of the Brillouin zone, i.e., ≈30 fs for RbMnF₃, excites oscillations of spin correlations with the largest amplitude (see Ref. 34 for details). This contrasts with the optimal excitation of spin waves in an antiferromagnet at the center of the Brillouin zone, which simply requires the shortest laser pulses.³⁶ Contrary to models in terms of M and L, the spin correlation function is capable of describing an adequate response to an anisotropic perturbation of short-scale spin–spin exchange coupling by a polarized laser pulse and also reveals the effect of the latter on the optical properties of the medium. An important outcome of the developed model is the intuitive analytical formula for time-dependent spin dynamics as well as polarization dependences of optical signals in pump-probe experiments, unavailable in earlier reported considerations of spin dynamics at the edge of Brillouin zone.^20,33,34 We note that regimes of ultrafast spin dynamics beyond precession of L and M are being also studied by atomistic^39,40 and multiscale simulation,⁴¹ which, however, focus on strongly dissipative light–matter interactions and still yield longitudinal and transverse dynamics of M and L. Our experimental and theoretical findings open up novel perspectives for ultrafast antiferromagnetic spintronics and magnonics governed by laws different from those established for low-energy spin waves.

See the supplementary material for a detailed derivation of the equation of motion for spin correlations and the equation for ɛ^xx − ɛ^yy, and derive the polarization dependences directly from these equations. We also discuss the key problems one faces when describing the observed dynamics using a Néel vector L.

We are grateful to A. K. Zvezdin, V. N. Gridnev, A. I. Nikitchenko, D. Bossini, H. Hedayat, and P. H. M. van Loosdrecht for fruitful discussions and S. Semin and C. Berkhout for technical support. J.H.M and D.I.K. acknowledge funding from the European Research Council under ERC Grant Agreement No. 856538 (3D-MAGiC) and J.H.M. acknowledges the Horizon Europe Project No. 101070290 (NIMFEA). D.A. acknowledges the support of ERC Grant No. 101078206 ASTRAL. The work of T.T.G. was funded by the European Union’s Horizon 2020 Research and Innovation Program under Marie Skłodowska-Curie Grant Agreement No. 861300 (COMRAD). R.M.D. acknowledges the support of RSF (Grant No. 22-72-00025). A.M.K. acknowledges support from RFBR (Grant No. 19-52-12065).

The authors have no conflicts to disclose.

The authors declare that this work has been published as a result of peer-to-peer scientific collaboration between researchers. The provided affiliations represent the actual addresses of the authors in agreement with their digital identifier (ORCID) and cannot be considered as a formal collaboration between the aforementioned institutions.

F. Formisano: Investigation (equal); Writing – review & editing (supporting). T. T. Gareev: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – review & editing (equal). D. I. Khusyainov: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (supporting); Writing – review & editing (supporting). A. E. Fedianin: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). R. M. Dubrovin: Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). P. P. Syrnikov: Resources (equal). D. Afanasiev: Methodology (equal); Supervision (lead); Writing – review & editing (equal). R. V. Pisarev: Resources (lead); Writing – review & editing (equal). A. M. Kalashnikova: Conceptualization (lead); Funding acquisition (lead); Writing – original draft (lead); Writing – review & editing (lead). J. H. Mentink: Conceptualization (lead); Funding acquisition (lead); Methodology (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead). A. V. Kimel: Conceptualization (lead); Funding acquisition (lead); Resources (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.