Spin wavepackets in the Kagome ferromagnet Fe3Sn2: Propagation and precursors

Significance Wavepackets of magnetization in magnetically ordered materials have emerged as a potential means to shuttle quantum information over large distances. A particularly promising platform is quasi-two-dimensional magnets in which the spins within each atomic plane are parallel but interplane order can be ferromagnetic or antiferromagnetic. In this work, we use ultrashort light pulses to generate spin wavepackets in a kagome-layered ferromagnet and to follow their subsequent motion. A significant result is that the arrival of magnetization at a remote location occurs in a time far shorter than expected from the spin wave velocity. We show that this “precursor” originates from the long-range magnetic dipole interaction. Related effects may have far-reaching consequences toward realizing long-range transport of spin information.


INTRODUCTION
Harnessing electron spin is one of the central goals of condensed matter physics.A particularly exciting direction is the coupling of spin to charge and lattice degrees of freedom to provide interconnections in hybrid quantum systems.To this end, it is essential to understand and control the generation, propagation, and detection of spin information.
Recent progress in magnetically ordered systems has shown promise in using propagating spin waves -collective excitations of the electron spins -to transport information over large distances [1][2][3][4][5].Increasingly, attention has focused on quasi-two dimensional (2D) layered magnets in which spins within each plane are parallel but the interplane order can be ferromagnetic [6], antiferromagnetic [7], or even helical [8].
An important subset of such systems is "easy-plane" magnets in which spins are oriented parallel to the planes but without a preferred direction within the plane.As a result of the symmetry with respect to in-plane spin rotation, the out-of-plane, or z component of the magnetization, M z , is a conserved quantity.Theoretically, M z can exhibit ballistic, diffusive, hydrodynamic, or even superfluid regimes of transport [9].However, in real 2D easy plane magnets this rotational symmetry is broken, although weakly, by the anisotropy of the underlying lattice.This fact has driven theoretical studies of the consequences of rotational symmetry breaking and approaches to mitigating its effects [10,11].
At low temperatures, Fe 3 Sn 2 exemplifies an easy-plane system of the class introduced above, in which the spins experience weak anisotropy resulting from the discrete 3-fold ro-tational symmetry of the rhombohedrally stacked Kagome lattice [12][13][14].In this work we study the propagation of spin wavepackets in Fe 3 Sn 2 , using temporal and spatially resolved optical techniques to probe their amplitude, frequency, and velocity.In our pump/probe measurement scheme, the pump pulse excites a spin wavepacket whose propagation is detected by a time-delayed and spatially-separated probe pulse through the magneto-optic Kerr effect (MOKE) [15,16] or optical birefringence [17].The range of wavevectors that comprise the spin wavepacket is determined by the Fourier transform of the real space excitation density, which is typically Gaussian.
Since the size of the focused laser spot is diffraction limited, the excited wavevectors are typically within the range of inverse micrometers (µm −1 ).In this long wavelength regime, the propagation of spin is dominated by magnetic dipole interactions, drastically altering the properties that arise from short-ranged exchange interactions alone [18,19].Excitations in this regime are referred to as magnetostatic spin waves (MSWs) although they are fully dynamic; the term arises because their dispersion relations can be obtained within the magnetostatic approximation, ∇ × H = 0, which is valid because spin wave velocities are much smaller than the speed of light.
Given the long-range nature of the dipole interaction, MSWs are particularly sensitive to both the shape of the medium and magnetic anisotropy.Damon and Eshbach (DE) [18] obtained MSW dispersion relations for a magnetic slab with uniaxial anisotropy, associated with either an easy axis or applied magnetic field.However, as we demonstrate below, spin wavepacket propagation in Fe 3 Sn 2 shows novel properties that cannot be described by the DE relations, including the remarkable observation that spin excitations can be detected remotely at a time much shorter than would be inferred from the spin wave velocity.In the theoretical component our study, we use both analytical calculations and numerical modeling to show that these effects are accounted by extending the DE formalism to the easy plane systems of current interest.Although the theory presented below assumes ferromagnetic order between the planes, as in Fe 3 Sn 2 , it applies to antiferromagnetic order as well.For example, the antiferromagnetic version of the theory provides a quantitative explanation for the recently discovered surprising properties of spin wavepacket propagation in the 2D van der Waals antiferromagnet CrSBr [20].

Magnetic field dependence of spin wave frequency
Prior to measurements of spin transport, the anisotropy parameters of Fe 3 Sn 2 were determined using the time-resolved magneto-optic Kerr effect (TR-MOKE).In this method, a short (∼100 fs) duration laser pulse induces a transient misalignment between the magnetization, M, and the effective anisotropy field, H eff .The resulting torque causes M to precess, as illustrated in Fig. 1(a).The precession leads to oscillations of the component of magnetization parallel to the optic axis, M z , which are detected via the polar Kerr effect.[15,16,21].
Figures 1(b) and 1(c) show oscillations of M z as detected by the TR-MOKE for several magnetic fields applied in the z direction.Fig. 1(d) displays the Fourier transform of the oscillations in the frequency-magnetic field plane; the dashed line is a fit to a model described below.This dependence of spin wave (SW) frequency on field is characteristic of a ferromagnet whose biaxial anisotropy can be described by the free energy , where K x and K z are the in-and out-of-plane anisotropy energies (K x , K z > 0) and x is a preferred magnetization direction within the plane.The origin of the inplane anisotropy is discussed in Supporting Information Sections I and II, which present a microscopic model whose low-temperature, broken-symmetry phase is described by F A for small fluctuations of the ferromagnetic order parameter.The theoretically predicted dependence of SW frequency on magnetic field, ν(H z ), for a biaxial ferromagnet is [22,23], where H s is the saturation field along the z direction, M s is the saturation magnetization, and γ is the gyromagnetic ratio.Eq. 1 accurately describes the full field dependence of the TR-MOKE frequencies observed in our experiment (see Supporting Information Section V).
The fit (dashed line in Fig. 1d) yields K x ≈ 1.76 × 10 4 J/m 3 and K z ≈ 2.26 × 10 5 J/m 3 , consistent with the picture of weak anisotropy within an easy plane.We now describe extending time-resolved measurements to the spatial domain.A simplified layout of the setup for TR-MOKE microscopy is shown in Fig. 2(a).The 4f optical system equipped with 2-axis galvo-driven mirrors enables continuous scanning of the pump focus in two dimensions while the location of the probe is fixed [24].Spin waves photoexcited in one location can be probed remotely at a subsequent time, enabling an all-optical ultrafast investigation of SW transport with micron-scale spatial resolution and sub-microradian polarization sensitivity.proportion to e −(∆x/σ) 2 , where σ ∼ 6µm.In this regime, the decay of the amplitude reflects the spatial overlap of pump and probe foci (with full width at half-maximum (FWHM) spot sizes of 6 and 5 µm, respectively) as would be expected in the absence of propagation.
However, for larger ∆x spin propagation becomes evident; for ∆x > 10µm the TR-MOKE amplitude deviates from a Gaussian and at ∆x = 20µm is four orders of magnitude larger than can be accounted for by spatial overlap.
The distinction between the overlap and propagation regimes is also seen by normalizing the TR-MOKE traces to the amplitude at zero separation.that spin waves have reached this distance occurs at ≈100 ps, from which we estimate an effective velocity of ≈ 2 × 10 7 cm/s.This velocity is six orders of magnitude larger than the group velocity inferred from neutron scattering measurements [25].In the following section we show that discrepancy is resolved by considering wavepacket propagation in the magnetostatic regime.

DISCUSSION
As mentioned in the introduction, spin wavepacket propagation in Fe 3 Sn 2 cannot be described by the DE dispersion relations for either surface or bulk modes.For example the DE surface mode is nonreciprocal, with a single direction of propagation that is reversed for the two opposing surfaces.Instead, we observe reciprocal propagation, that is symmetric with respect to wavevector k → −k.The volume modes, although reciprocal, propagate only along one axis, whereas we observe propagating modes along two principal axes in the plane.Furthermore, the bidirectional DE volume mode is "backward moving" in the sense that its phase and group velocities are opposite, whereas we find the two principal axes of propagation exhibit forward and backward modes, respectively.As we show below, extending the DE calculation to nearly easy-plane systems accounts for the novel features observed in our spin transport measurements.

Magnetostatic spin waves under biaxial anisotropy
We consider a geometry with the equilibrium magnetization in the plane and parallel to one of the easy axes [26,27].Maxwell's equations in the magnetostatic regime, ∇ • B = ∇ × H = 0, together with the Landau-Lifshitz equation, where H eff is the sum of the anisotropy field and the dynamical field h, form a closed set that yield the normal modes of magnetization in the long-wavelength regime.To illustrate the resulting MSW dispersion, Fig. 4

Spin wavepacket propagation
We turn next to the dynamics of wavepackets whose motion is determined by the dispersion relation illustrated in Fig. 4. The primary goal is to understand how spin information Conventionally, the slope of a fit to these points yields the group velocity, v g .From this perspective, the data are quite puzzling, as v g appears to increase with time, reaching anomalous value, ≈ 2 × 10 6 cm/s, much larger that expected for spin waves.Finally, Fig. with,

Physical origin of anomalous propagation
In the previous section we showed that spin wavepacket dynamics in Fe 3 Sn 2 can be quantitatively modeled by the MSW dispersion relations for a biaxial ferromagnet.In this section offer a physical picture that underlies the most puzzling feature of the wavepacket propagation -apparent velocity in excess of the expected SW velocity.Essentially, the seemingly anomalous behavior is a consequence of a breakdown of the group velocity description that occurs when a dispersion relation is highly structured within the range of wavevectors that comprise the packet.In the following, we show that dynamics in this regime can lead to early arrival times at remote locations, which we refer to as spin wave precursors.
To illustrate the origin of spin wavepacket precursors, consider the V-shaped dispersion relation for propagation in the y direction shown in Fig. 5(d).In this approximation to the actual relation (Fig. 4(b)), SWs propagate with constant velocity for k y < 2k z , and do not propagate for k y > 2k z .The precursor effects arise from SW modes in which k z σ is small, such that the Gaussian distribution of photoexcited wavevectors (green line in Fig. 5(d)) spans both propagating and non-propagating regimes.
The time-evolution of the wavepacket is given by summing the contributions from the two regimes, where ω 0 is the frequency at k y = 0 and v is the slope of the V-shaped region.Fig. 5(e) shows the propagating and non-propagating terms in Eq. 6 evaluated at t = 0 (red and blue lines, respectively), together with their sum (green line).The individual terms in Eq.
6 are oscillatory with a slowly decaying envelope, as expected for the Fourier transform of a sharply truncated Gaussian.Notice that the oscillations cancel out under summation, yielding the initial Gaussian wavepacket.However, for t > 0 the propagating component moves away from the origin at velocity v while the nonpropagating component remains stationary, disrupting the initial cancellation of the two components.This effect manifests as appearance of oscillations in magnetization at large distances within a short time frame.In this simplified picture, a spin wave precursor can be seen at arbitrarily large distances within a time of order of the precession period.In reality, the range of detection will be limited by the rounding of the dispersion neglected in our V-shape approximation; nevertheless precursors will appear on time scales that are not set by the SW velocity.

CONCLUSION AND OUTLOOK
We have shown that spin waves in Fe 3 Sn 2 can be optically excited, propagated, and de- As the profile of the propagating modes evolves for t > 0, the oscillations associated with the nonpropagating modes no longer cancel.The total Kerr rotation shown in green reveals evidence of propagation even at separations greater than v g t.

Crystal growth
Single crystals of Fe3Sn2 were grown using a Chemical Vapor Transport method with conditions outlined in Ref. [13].The resulting crystals tend to be hexagonal thin plates, and optical measurements were performed on as-grown (001) surfaces.

Field dependence measurements
The time-resolved magneto-optic Kerr effect (tr-MOKE) measurements with an out-ofplane magnetic field were performed with 1560 nm pump and 780 nm probe laser pulses generated from a Menlo C-Fiber erbium fiber oscillator operating at a repetition rate of 100 MHz.The pump and probe powers were set to 20 mW and 0.1 mW, and focused onto the sample surface with approximate spot sizes of 20 µm and 6 µm, respectively, using an objective lens with a numerical aperture (N.A.) of 0. the magnon gap at Γ is a six-spin term ∼ (S + ) 6 + h.c..In Fe 3 Sn 2 , the simplest realization of this term is a 3-site ring exchange interaction: where i, j, k denotes nearest neighbor triplets i, j, k on the same triangle.
To summarize, our minimal model to describe the spin dynamics of Fe 3 Sn 2 is given as follows: where B z labels the Zeeman field applied along c−axis.At k = 0, with B z = 0, we can work out the magnon spectrum using a Holstein-Primakoff transformation: where b † i , b i are boson creation and annihilation operators, and S is the total spin on each site i.After making these substitutions, isolating quadratic terms, and converting to momentum space, we can write the boson BdG Hamiltonian as where and there are m = 3 spins in each unit cell.At k = 0, we have the simple form where where we define δ ≡ ∆ − 1.To find the magnon spectrum, we diagonalize the matrix σ z H b , where σ z acts on the ( b, b † ) space.We find that two of the bands are degenerate at k = 0 For small values K J , |δ| 1, the non-degenerate band is lower in energy, with energy given by where K ≡ KS 3 (2S − 1).The experimentally determined spin stiffness of Fe where a = 5.34 Å is the the lattice constant.Given the gap E 1 ≈ 8 GHz ≈ 33 µeV, one can estimate the exchange coupling to be where we have set S ∼ O(1) in the estimation.This means a very small and realistic ring exchange coupling can already induce the 0.03 meV magnon gap at the zone center.

II. FREE ENERGY ANALYSIS
Consider a classical ferromagnetic ground state with magnetization vector is denoted as M = (M x , M y , M z ).The free energy of the anisotropy F A can be written as where K x is the in-plane anisotropy energy, K z is the out-of-plane anisotropy energy (K x , K z > 0) and M s is the saturation magnetization.The effective field induced by the anisotropy energy can be derived as In addition to H A , the effective field H ef f includes the dynamic demagnetizing field h, which is the key to consider the magnetic dipole-dipole interaction.Therefore, our goal is to solve the Landau-Lifshitz equation below to calculate the magnon dispersion, where γ is the gyromagnetic ratio.We assume that the equilibrium magnetization is along the x-direction and consider an infinitely large sample, so that the normal modes have the form of plane waves.We have M y = m y (r, t) = m y e i(k•r−ωt) , M z = m z (r, t) = m z e i(k•r−ωt) , h x (r, t) = h x e i(k•r−ωt) , h y (r, t) = h y e i(k•r−ωt) , h z (r, t) = h z e i(k•r−ωt) .

(S34)
Since m y,z and h x,y,z are the dynamic magnetization and dynamic magnetic fields, m y,z , h x,y,z << M s .Substituting Eq.S34 into Eq.S33, we get the relation between m y,z and h y,z . where Noting the relation ∇ × h = 0, we can consider the dynamic magnetic field as the gradient of a scalar potential ψ, ψ = ae i(k•r−ωt) , h = ∇ψ = ikae i(k•r−ωt) .

(S37)
Taking the dipole-dipole interaction into account, we use the Gauss's Law of the Maxwell's equations where b = h + 4πM .Using the relation in Eq.S35, we have where ω M = 4πγM s .Then Eq.S38 becomes Recalling that ψ = ae i(k•r−ωt) , (S41) we have By solving Eq.S42, we get the dispersion relation of the dipolar magnon mode, The dipolar magnon dispersion relations at selected values of k z are plotted in Fig. S2

V. THE DIPOLAR MAGNON DISPERSION IN AN EXTERNAL FIELD
We now include an external field applied along the direction of the hard axis (z-axis).The free energy becomes,

5 K
FIG. 1.(a) Illustration of magnetization spiraling to align with H eff .(b,c) Kerr rotation as function of pump-probe delay shown for applied magnetic fields ≤ 0.75 and > 0.75 T, respectively.The curves are offset for clarity.(d) Amplitude of Fourier transforms of the time series plotted in (b) and (c) shown in the frequency-field plane.The dashed line indicates the fit to Eq. 1.All data were taken at T = 2.5 K

FIG. 2 .
FIG. 2. Time-resolved MOKE microscopy (a) Overview of the experimental setup.The 2D galvo mirrors and the 4f optical geometry enable scanning of the pump laser beam.(b)-(i) Snapshots of 2D MOKE maps taken at several pump-probe time delays.For time delay t ≥ 180 ps (f-i), the propagation is clearly anisotropic, with contrasting properties along two principal axis directions.All measurements were performed at T = 2.5 K with an out-of-plane field of 0.5 T.

Fig. 3 (
c) shows the normalized signals for ∆x < 10µm, which is in the Gaussian regime of Fig. 3(b).For these separations the envelope of the TR-MOKE oscillations decays monotonically with increasing time, consistent with a simple damped response.However, at separations greater than 10µm, shown in Fig. 3(d), the envelope peaks at a nonzero time delay, as expected for a propagating wavepacket.Focusing on the arrival time of the wavepacket at the largest measured separation of 22µm reveals another surprising feature.Notice that the first clear indication

FIG. 3 .
FIG. 3. (a) TR-MOKE traces at different values of spatial separation (∆x) between the pump and probe beams.(b) The log of the amplitude of the Fourier transform of the data shown in (a) is plotted vs. (∆x) 2 (black dots).The red line is the rate of decrease expected in the absence of propagation.(c,d) Normalized TR-MOKE traces at separations for ∆x < 10µm and ∆x > 10µm, respectively, illustrating the change in the envelope function from exponential to Gaussian.All measurements were taken at T = 2.5 K under an out-of-plane field of 0.5 T.
(a)  shows the calculated spin wave frequency in the k x , k y plane for fixed k z =1 µm −1 .Line cuts through this plane defined by k x = 0 (purple) and k y = 0 (orange) plotted in Fig.4(b) show forward propagation along the y direction and backward along x, with a saddle point at q = 0.When K z > K x , as in Fe 3 Sn 2 , the velocity is larger along k y .This dispersion relation was also reproduced through micromagnetic simulations.These predictions are unique to biaxial ferromagnets and clearly distinct from the uniaxial (DE) limit, in which there are no forward propagating reciprocal modes (see Supporting Information Sections IV-VI for the calculations and numerical simulation of the MSW dispersion relations).The prediction of a saddle dispersion relation was tested by measuring the TR-MOKE oscillations as a function of pump/probe separation along the two principal axes of propagation identified in the maps shown in Fig.2.The results are presented in Figs.4(c) and 4(d) as color plots in the time-separation plane.The slope of the lines of constant phase distinguishes forward vs. backward propagating modes.In agreement with our theoretical prediction for the biaxial ferromagnet, modes with wavevector perpendicular to M are forward propagating, and backward propagating for wavevectors parallel to M.

FIG. 4 .
FIG. 4. Magnetostatic waves (MSWs) under biaxial anisotropy.(a) Three dimensional representation of the calculated MSW dispersion of Fe 3 Sn 2 as a function of k x and k y evaluated at k z = 1 µm −1 .A saddle point can be observed at the origin.(b) Frequency-momentum cuts at k x = 0 (purple) and k y = 0 (orange) illustrating forward propagation along k y (purple) and backward propagation along k x (orange).(c,d) Plots of the TR-MOKE ampitude in the ∆x, t plane measured along the two principal axes of propagation show forward and backward propagation, respectively.

5
(c) presents a zoomed in view of the wavepacket amplitude vs. separation, now on a double logarithmic plot.Below we show that the MSW dispersion relation, ω(k), in biaxial magnets successfully explains the anomalous wavepacket propagation in Fe 3 Sn 2 .Crucially for the interpretation of our experiments, photoexcitation launches a coherent spin wavepacket, comprised of a Gaussian distribution of wavevectors that are initially in phase.The time-and position-dependent magnetization detected by TR-MOKE can be calculated using the following relation: where g(k) is the Fourier transform of the initial perturbation generated by the pump pulse and m(k) is the normal mode eigenvector.The radius of the focused laser beam, σ, and the anisotropy parameters are determined from independent measurements.The only adjustable parameters in the theory are the damping constant, α, and the effective penetration depth, δ p , of the perturbation that induces the subsequent precessional motion.Parameter values, δ p = 230 nm and α = 2.7 × 10 9 s −1 were chosen to achieve the best fit (solid red line) to the amplitude vs. distance data shown in Fig.5(c).The same parameters accurately reproduce the anomalous wavepacket position vs. time data as well (red line in Fig.5(b)), adding additional support for our theoretical model (see Supporting Information Section VII for details).

FIG. 5 .
FIG. 5. (a) Normalized TR-MOKE amplitude vs. t for values of ∆x > 10µm with arrowheads indicating the center of the wavepacket.(b) The solid circles show the wavepacket center as a function of time.The red line is a fit based on the calculated MSW dispersion relation.(c) A double logarithmic plot of the amplitude of the wavepacket vs. pump-probe separation (solid circles) and the fit (red line) using the same parameters as in (b).(d-f) Illustration of the physical origin of the wavepacket precursor.(d) Shown as a green line is the Gaussian distribution of wavevectors excited by the pump beam.The purple line is an approximation to the MSW dispersion for a value of k z that is within the range of excited in-plane wavevectors.The regimes with group velocity v g > 0 and v g = 0 are indicated.(e) Red and blue lines show the contributions to the total Kerr rotation from the propagating and nonpropagating modes, respectively, evaluated at t = 0.As expected, their sum yields a Gaussian profile corresponding to the initial photoexcited state.(f)

25 .
The transient changes in Kerr rotation values were subsequently measured with a balanced photodetection scheme and a lock-in amplifier.The pump laser pulses were modulated at 100 kHz with a photo-elastic modulator (PEM).Propagation measurementsThe non-local propagation experiments were carried out with 514 nm pump and 633 nm probe pulses generated from the ORPHEUS-TWINS optical parametric amplifiers pumped by the Light Conversion CARBIDE Yb-KGW laser amplifier operating at the repetition rate of 600 kHz.Both beams were focused onto the sample sample surface with approximate spot sizes of 6 µm and 5 µm, respectively, with incident laser powers fixed at 30 µW.The position of the pump focus was scanned by adjusting the voltage applied to the 2-axis galvanometerdriven mirrors, which are located at a distance 4f (f = 50 cm) before the entrance aperture of the final objective lens (N.A. = 0.25).A pair of telescope lenses with focal lengths of f are placed equidistant from the galvo mirrors and the objective so that the laser beam steered from the galvo mirrors forms a one-to-one image at the entrance of the objective lens.The pump laser pulses were modulated at 100 kHz with a PEM.

Figure S1 .
Figure S1.Angular dependence of the spin wave propagation.(a) Overview of the experimental setup where the orientation angle between the pump and probe beams is being continuously varied.(b)-(d) Angle-dependent tr-MOKE measurements taken at pump-probe separation distances of 12.8, 14.4, and 16 µm, respectively.Different colors indicate various time delays between the pump and probe pulses.

Figure S2 .
Figure S2.The k z dependence of the dipolar magnon dispersion.Line cuts taken through the plane defined by (a) k y = 0 and (b) k x = 0 are shown for selected values of k z .