Non-equilibrium dynamics of spin-lattice coupling

Quantifying the dynamics of normal modes and how they interact with other excitations is of central importance in condensed matter. Spin-lattice coupling is relevant to several sub-fields of condensed matter physics; examples include spintronics, high-Tc superconductivity, and topological materials. However, experimental approaches that can directly measure it are rare and incomplete. Here we use time-resolved X-ray diffraction to directly access the ultrafast motion of atoms and spins following the coherent excitation of an electromagnon in a multiferroic hexaferrite. One striking outcome is the different phase shifts relative to the driving field of the two different components. This phase shift provides insight into the excitation process of such a coupled mode. This direct observation of combined lattice and magnetization dynamics paves the way to access the mode-selective spin-lattice coupling strength, which remains a missing fundamental parameter for ultrafast control of magnetism and is relevant to a wide variety of materials.


Introduction
Crystals with magnetic order have dynamics governed largely by normal modes of lattice and spin displacement, giving rise to quasiparticles such as phonons and magnons.The frequencies of the modes are defined by the microscopic interaction among individual atoms and among the magnetic moments localized at magnetic ions.These excitations are commonly studied by spectroscopic techniques such as inelastic neutron scatterings, X-ray scatterings, and Raman spectroscopy.In some cases, direct measurements of coherent excitations in the time domain have become possible, for example by performing pump-probe measurements of coherent vibrational and spin dynamics [1,2].
The coupling between spins and lattice is a fundamental interaction important to a wide variety of magnetic and electronic phenomena.Examples are ultrafast energy and angular-momentum transfer [3] and electric control of magnetism [4], relating to efficient energy transfer and storage application.This coupling can also lead to hybrid normal modes that involve both lattice and spin dynamics.
In this study, we directly observe how the spins and lattice interact as components of a hybrid normal mode.We focus on a specific multiferroic where the spin-lattice coupling is the origin of the magnetoelectric effect and responsible for non-reciprocal transport [5].The breaking of inversion symmetry by magnetic order can cause magnetic excitations to acquire an additional polar phononic character, making them susceptible to the electric field component of light.The characteristic hybrid mode that arises from the first-order coupling between magnetism and ferroelectricity is called electromagnons [6] and provides a platform to us to investigate spin-lattice coupling in this material.
Electromagnons were originally envisioned as excitations arising from the coupling between magnetic order and the crystal lattice that gives rise to the equilibrium electric polarization [7].We refer to this type of electromagnons as type I.While type I electromagnons have been reported [8], for many materials magnetostrictive coupling to a polar lattice distortion gives rise to electromagnons with much larger oscillator strengths [9][10][11].In these electromagnons (we refer to them as type II) the polar lattice distortion modifies exchange interactions, resulting in a magnetic response.Coherent excitations of these hybrid modes with pulsed electric fields hold great potential to manipulate magnetism on an ultrafast timescale, as theoretically predicted for TbMnO3 [12].While the spin dynamics of coherently excited electromagnons have been observed by time-resolved magnetic diffraction [13][14][15], the underlying structural dynamics of electromagnons have not yet been directly observed in any material so far.
The Y-type hexaferrites (Ba,Sr)2Me2(Fe,Al)12O22 (Me: transition metal ion) belong to the family of magnetoelectric multiferroics with space group R-3m and a large lattice parameter along [001] (» 43.3 Å), as shown in Fig. 1a.Their magnetic structure is commonly described as a magnetic block structure, where each magnetic block, termed S or L, contains a certain number of collinearly coupled Fe magnetic moments [16].The Fe-O-Fe bond angles at the boundary between adjacent magnetic blocks create magnetic frustration.As a result of the magnetic frustration, the Y-type hexaferrite investigated here exhibits the transverseconical magnetic structure, as displayed in Fig. 1b.The magnetic structure can be described as a combination of a ferrimagnetic component and a spin-spiral component.While the ferrimagnetic component hosts net magnetization, the spin-spiral component gives rise to electric polarization via static spin-lattice interaction [16].This makes the transverse-conical phase multiferroic.
A clear electric-dipole active resonance in the THz absorption spectra has been reported in Y-type hexaferrites for electric fields (ETHz) parallel to [001] [17,18].Based on a detailed investigation of the temperature, magnetic-field, and ETHz-direction dependences of the THz optical properties [17,18], this mode has been attributed to an electromagnon mode.Out-of-phase oscillation of the magnetic moments towards [001] in the magnetic blocks can result in a transient electric dipole moment along [001] via magnetostriction, as schematically illustrated in Fig. 1c [17].Resonant excitation of electromagnons by an intense THz pulse is known as an effective pathway to create large amplitude dynamic modulations of magnetic moments [13].The large resonator strength found in Y-type hexaferrites implies a strong spin-lattice coupling, which is promising for possible ultrafast control of magnetism.Although the theory of these electromagnons is well developed, experimental observation of the vibrational component of electromagnons is still missing.The measurement of this structural component in combination with the magnetic response is of fundamental importance for experimentally demonstrating how electromagnons are excited following a THz pulse in a multiferroic.Moreover, it has the potential to directly determine the magnetoelectric (spin-lattice) coupling strength of the excitation.
Here we utilize time-resolved hard X-ray diffraction (tr-XRD) and time-resolved resonant soft X-ray magnetic diffraction (tr-RSXD) with few-cycle phase-stable THz excitations to observe the lattice and sublattice magnetization dynamics of electromagnons, respectively, in the multiferroic Y-type hexaferrite Ba1.3Sr0.7CoZnFe11AlO22.Clear oscillations in the time dependence of the (0 0 24) reflection intensity show the lattice component of this mode, whereas the oscillations of the (0 0 4.5) magnetic reflection intensity give a direct measure of the induced spin motion.Observed redshifts concomitant with increased damping at elevated temperatures confirm the identification of the oscillations as being due to the electromagnon.A direct comparison of magnetic and lattice responses opens the door for accessing mode-selective spin-lattice interactions important for ultrafast control of materials, and allows us to discuss how the two characters couple on ultrafast timescales.The white region is a frequency range where no transmission through the sample is detected because of strong absorption.The crystal structure was drawn by VESTA [41].

Lattice dynamics
THz time-domain spectroscopy (THz-TDS) data on the Y-type hexaferrite show a clear divergent feature centered at ~1.7 THz (T = 20 K) in the imaginary part of the refractive index (see Figs. 1d and 1e).Upon heating, the mode shows a red shift concomitant with a weakening of the resonator strength.These results are consistent with the previous literature identifying the mode as an electromagnon [17,18].Figure 2a shows a time trace of the electric field of the incident THz pulse (ETHz) in the tr-XRD experiment, whose direction was rotated to maximize the component along [001] (see Methods).The corresponding time trace of the (0 0 24) reflection intensity taken at a sample temperature of 20 K is shown in Fig. 2c.The figure also shows measurements of the time-dependent X-ray diffraction for opposite phases of the THz pulse (+ETHz and -ETHz).Figures 2b and 2d show the corresponding Fast Fourier Transformation (FFT) amplitudes of the incident THz pulse and the (0 0 24) reflection intensity, respectively.As described in Methods, the photon energy used for the experiment is 7.5 keV, slightly above the Fe K edge, where the large imaginary part of the Xray scattering factor of Fe enables us to detect polar motions.Clear oscillations are observed in the diffraction intensities.These dynamics change sign when reversing the ETHz direction, unequivocally indicating coherent dipole-active lattice vibrations.The FFT of the time traces reveals two peaks at frequencies 1.7 THz and 3.6 THz, with the latter having an amplitude about five times larger than the former.The lower frequency matches that of the [010] [001] n electromagnon resonance, while the higher frequency corresponds to an infrared-active optical phonon mode [16].Note that the spectral power of the incident THz pulse at 3.6 THz is approximately an order of magnitude weaker than that around 1.7 THz, as shown in Fig. 2b.To exclude that the observation of the higher-frequency mode represents a phonon upconversion process caused by non-linear phonon interactions as observed in SrTiO3 [19], a low-pass filter has been inserted to fully suppress the weak high-frequency component in the THz pulse (see Methods for detail and Fig. 2b for spectrum with the low-pass filter).The absence of the higher-frequency component in the tr-XRD data taken with the filter (Fig. 2e) and in the FFT of the diffraction intensities (Fig. 2f) fully supports that the higher-frequency mode is resonantly driven by the weak spectral weight around that frequency.As for the lowfrequency mode, it has a phase shift of π/2 compared to the THz pulse, consistent with resonantly driven lattice dynamics described by the Lorenz model (see Supplementary Information Sec.I).For increasing temperatures, the mode softens with increased damping (Fig. 2f) consistent with the THz-TDS data (Figs.1d and 1e).We infer from this that the measured structural dynamics of the mode at ~1.7 THz are the structural component of the electromagnon.
One striking feature of our results is the strong difference in the response of the XRD signal of the electromagnon mode when compared to that of the polar vibrational mode at 3.6 THz for data taken without the low-pass filter.As remarked above, even though the spectral power at 3.6 THz in the incident pulse is approximately an order of magnitude weaker than at 1.7 THz, the measured response at 3.6 THz is approximately five times stronger.Without knowing the eigenvector of each mode, it is at present not possible to make a quantitative link between the diffraction response and the atomic displacements associated with each mode.Under the assumption that the (0 0 24) reflection has a similar sensitivity to both modes, however, our results would suggest that a substantial amount of the absorbed THz energy at 1.7 THz is within the spin system.Note that the resonator strength of the electromagnon mode is distinctly larger than that of the optical phonon mode (see Fig. 1e), as can be seen from the previously reported THz reflectivity data taken at low temperatures [18].

Magnetic sublattice dynamics
Figures 3a and 3b show a time trace of the electric field of the incident THz pulse in the tr-RSXD experiment (3a) and its FFT spectrum (3b), respectively, which are similar but with a slightly weaker driving field strength compared to the tr-XRD experiment.The pump beam polarization was rotated to maximize the field component along [001], as for the tr-XRD experiments.The corresponding time traces of the antiferromagnetic (0 0 4.5) reflection intensity taken at the Fe L3-edge and at various temperatures are shown in Fig. 3c and their FFT spectra in Fig. 3d.This reflection probes the spin-spiral component of the magnetic order [20] (see Supplementary Information Sec.V for a discussion of other negligible contributions).The data reveal that intensity variations are about twice as large as those in the structural reflection indicating clear resonantly driven spin dynamics in the hexaferrite.The weakening and redshift for increasing temperature are in qualitative agreement with the behavior of the electromagnon.The obtained low-temperature frequency, however, seems slightly lower in the magnetic response than in the lattice response.Note that previously reported polarized inelastic neutron scattering for a Y-type hexaferrite [17] found magnetic excitation energies consistent with those of their/our THz-TDS data and tr-XRD data, confirming phononic and magnetic frequencies of the electromagnons being likely equal.However, due to the small applied magnetic field, the mode could split and could be composed of multiple modes.In addition, the spectral weight of the THz pulse is larger at the low energy range in the tr-RSXD experiments than in the tr-XRD experiments and the probe depth is significantly shorter (see Methods), both of which could slightly modify the response frequency.However, in strong contrast with the tr-XRD data, which uniquely sample lattice dynamics, the oscillations in the magnetic tr-RSXD data exhibit a π phase shift with respect to the THz pulse.

Discussion
A quantitative assessment of the spin-lattice interaction of the electromagnon requires knowledge of the vibrational and magnetic components in the excitation.For the relative atomic dynamics of the two modes observed in tr-XRD, we need knowledge of the eigenvector for the modes and/or measurements of additional diffraction peaks.Because of the complicated crystal structure, modeling the vibrational dynamics of the electromagnons in the Y-type hexaferrite on an atomic scale is rather challenging and beyond the scope of this study.However, the oxygen atoms positions located on the borders between the adjacent magnetic blocks are known to dominate magnetic frustration [16], which, in turn, stabilizes the transverse-conical phase shown in Fig. 1b.We expect that even small displacements of these oxygen atoms can significantly affect the magnetic structure.This is consistent with a weak structural electromagnon response in respect to the more general polar vibrational mode.By assuming that the oxygen motions along [001] (with respect to all the rest atoms) represent the vibrational part of the electromagnons in the Y-type hexaferrite, a ~0.32 pm displacement results in the 0.5% change in the (0 0 24) diffraction intensity observed at 20 K (see Supplementary Information Sec.II).This amount of displacement changes the bond angles dominating magnetic frustration by ±0.15º, which is large enough to significantly affect the exchange interactions among magnetic moments [21].
The proposed model of the spin dynamics of electromagnons for Y-type hexaferrites, shown in Fig. 1c, gives rise to oscillations of the (0 0 4.5) magnetic diffraction intensity via changes in the two spin canting angles of the L and S blocks as shown in Fig. S6.A quantitative estimation (see Supplementary Information Sec.VI) indicates spin excitation amplitudes in the order of those reported for TbMnO3 [13], where a rotation of the spincycloid plane of approximately 4% was observed.
From the obtained spin excitation amplitudes, we can estimate the energy required for the maximal moment deflection with an approximation of the effective magnetic Hamiltonian.The energy associated with the lattice excitation could in principle be obtained by a density-functional theory calculation.This, however, requires inclusion of the magnetoelectric coupling, which is extremely challenging.Instead, we can compare the magnetic energy with the absorbed THz energy per formula unit (see Supplementary Information Sec.VIII).Our simple estimations show that the magnetic system may acquire a significant fraction of the absorbed THz energy.This suggests that performing measurements like this study is a viable method for obtaining coupling coefficients of the spin-lattice coupling of the electromagnon resonance.
Another interesting observation is the difference in phase shift of the lattice vibration and the magnetic response relative both to each other and to the THz excitation.With respect to the THz pulse, the magnetic mode shows a π phase shift similar to that observed in TbMnO3 [13], whereas the phononic part shows a π/2 phase shift, consistent with a resonantly driven damped harmonic oscillator.In the TbMnO3 study, the phase shift was interpreted as a consequence of measuring components of the spin dynamics that comprise the conjugate momentum of an electric-dipole active normal mode consisting of both vibrational and spin components.In a simplified picture, the ETHz applies a direct force to the atoms, which causes changes in an effective magnetic field that then drives the spins.This is consistent with the "two-step" model of Ref. [11], where the ETHz alters the exchange coupling constants J, and the derivative of the Hamiltonian with respect to the spin then creates an effective field that moves the spins.Our observation of the phase shift of the atomic motion relative to the spins thus indicates that electromagnons in the Y-type hexaferrite are type II.
In summary, we have directly observed the coupled dynamics of a coherently driven electromagnon, meaning both the atomic and the spin motions.Although previous work has shown that large-scale coherent magnetic dynamics can be driven by resonant excitation of an electromagnon [13], direct measurements of both the magnetic and the atomic displacement components are needed to fully understand the underlying coupling mechanisms.These are essential for obtaining material control through electromagnon excitations.Our results show that it is possible to establish this important link by using ultrafast X-ray diffraction techniques, or in more general ultrafast X-ray techniques including X-ray magnetic circular and linear dichroism.A better understanding of the magnon-phonon coupling is beneficial for optimizing materials for ultrafast control of magnetism.More generally, the ability to selectively probe the components of hybrid excitations is useful in a variety of strong-correlated electron systems where it is important to disentangle strongly coupled degrees of freedom.This can potentially contribute to a better understanding of high-TC superconductivity [22], topological properties [23][24][25], spintronics [26,27], and quantum computing [28][29][30][31].

Methods
Time-resolved X-ray diffraction experiments.The tr-XRD experiments were performed at the Bernina endstation [32,33] of the ARAMIS branch of SwissFEL at the Paul Scherrer Institute (Switzerland).The tr-RSXD experiments were performed at the Furka endstation, which is a new endstation for time-resolved studies in condensed matters at the Athos branch of SwissFEL at Paul Scherrer Institute [34,35].
Few-cycle phase-stable THz pulses with a central frequency of ~1.7 THz were generated by optical rectification of near-infrared femtosecond laser pulses in an OH-1 crystal.The time-dependent electric field ETHz incident on the sample was measured by electro-optic sampling [36] using a [110] oriented GaP crystal with 100 µm thickness and 800 nm pulses copropagating with THz pulses.From the THz-induced polarization rotation angle or phase retardation of 800 nm pulses, one can obtain the absolute scale of ETHz.Figures 2a  and 3a show ETHz with the corresponding FFT spectra in Figs.2b and 3b.The peak ETHz strength is ~0.36 MV/cm and ~0.2 MV/cm for the experiments at Bernina and Furka, respectively.For the hard X-ray data shown in Figs.2e and 2f, the small high-frequency component above ~3 THz was suppressed by the insertion of a low-pass filter before the sample (compare orange and blue curves in Fig. 2b, spectra with and without the low-pass filter, respectively).The low-pass filter is a multi-mesh filter whose cutoff frequency is 3 THz and transmission below the frequency is more than 85% (QMC INSTRUMENTS Ltd.).
The X-ray photon energy for the hard X-ray experiment was ~7.5 keV, which is slightly above the Fe K edge, with a bandwidth of ~11 eV.For the soft X-ray data, the X-ray photon energy was set to 711 eV, the maxima of an Fe L3 edge X-ray absorption spectrum (see Fig. S4a in Supplementary Information Sec.IV).The energy resolution is approximately 1 eV.A Y-type hexaferrite single-crystal sample, grown by a flux method described in Ref. [37], with a surface parallel to (1 0 16) was used, resulting in an angle of ~30° with the (001) plane.The electric field polarization of the incident THz pulse was adjusted to maximize its component along [001] by rotating the polarization of the near-infrared femtosecond laser and the OH-1 crystal accordingly.A relatively large incident angle of the X-ray beam to the surface, ~58° for the hard X-ray experiment and ~92° for the soft X-ray experiment, results in a grazing exit angle of ~0.5° for the (0 0 24) reflection in the hard X-ray experiment while ~36° for the (0 0 4.5) reflection in the soft X-ray experiment.This leads to an excited probe depth of ~52 nm and ~13 nm for the two experiments (penetration depth of the THz pulse is estimated from Fig. 1d as ~10 µm at 20 K by assuming a single peak at the resonant frequency in the imaginary part of the refractive index).The sample is mounted together with a pair of permanent magnets (~0.1 T // [010]) to stabilize the multiferroic transverse-conical phase [38] and is cooled with a He-flow cryostat for all X-ray experiments.We used an invacuum Jungfrau pixel detector for the hard X-ray experiment to collect both a diffraction peak and fluorescence signals simultaneously and used the latter for the normalization of Xray intensity fluctuation of the incoming beam.On the other hand, for the soft X-ray experiment, we used an avalanche photodiode to collect diffraction intensities.The normalization was done by measuring the X-ray intensities of diffracted beams by transmission gratings placed upstream, using a photodiode [39].The two experimental geometries are illustrated in Extended Data Fig. 1.

Fig. 1 |
Fig. 1 | Basic structures/characterization of Y-type hexaferrite.a, Crystal structure and b, magnetic structure of the Y-type hexaferrite Ba1.3Sr0.7CoZnFe11AlO22.The magnetic structure is depicted by the representative magnetic moments of an S block and L block [µS (blue arrow) and µL (red arrow), respectively], which alternatively stack along [001].c, Proposed spin dynamics of the electromagnon in a Y-type hexaferrite [17]: snapshots of the dynamics at different times, t1 (phase 0) and t2 (phase π).A pair of the same adjacent family of magnetic moments (µS1 and µS2, and µL1 and µL2) show anti-phase oscillations towards [001].Such oscillations give rise to transient electric dipole moments along [001] through magnetostriction.d, Temperature dependence of the real part, n, and e, the imaginary part, k, of the refractive index n+ik in the THz range, measured by THz-TDS.The white region is a frequency range where no transmission through the sample is detected because of strong absorption.The crystal structure was drawn by VESTA[41].

Fig. 2 |
Fig. 2 | tr-XRD signals.a, The incident THz pump pulse measured with electro-optic sampling at the sample position (see Methods for detail).b, THz pulse spectra, where blue and orange curves indicate data without and with the low-pass filter, respectively.c, The (0 0 24) XRD intensity at 20 K as a function of pump-probe delay.The blue data show the XRD response for +ETHz, while the orange data show the response when inverting the phase of the THz electric field (-ETHz).d, The FFT spectrum of the diffraction response.e, Temperature dependence of the time-dependent diffraction intensity after insertion of a low-pass filter that cuts off components above 3 THz.f, The FFT spectra of e. Dashed lines in a and e highlight the phase shift of π/2 between the incident THz pulse and the (0 0 24) XRD intensity.

Fig. 3 |
Fig. 3 | tr-RSXD magnetic scattering signals.a, The incident THz pulse measured with electro-optic sampling at the sample position.b, THz pulse spectrum.c, The antiferromagnetic (0 0 4.5) diffraction intensity at various temperatures as a function of pump-probe delay.d, The FFT spectra of c. Dashed lines in a and c are guides for the eyes to visualize the phase shift (of π) between the incident THz pulse and the magnetic (0 0 4.5) diffraction intensity.

Fig. 1 |
Schematic view of the experimental setups for the timeresolved X-ray experiments.a, For tr-XRD and b, for tr-RSXD.The [001] direction has an angle of ~30° from the surface normal n.