Changmin Lee,Yue Sun,Linda Ye,Sumedh Rathi,Kevin Wang,Yuan-Ming Lu,Joel Moore,Joseph G. Checkelsky,Joseph Orenstein
Spin wavepackets in the Kagome ferromagnet Fe3Sn2: Propagation and precursors
spin transport,time-resolved Kerr micrsocopy,magnetostatic spin waves
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.
Spin wavepackets in the Kagome ferromagnet Fe3Sn2: Propagation and precursors

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.

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 (1–5). 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, Mz, is a conserved quantity. Theoretically, Mz 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, Fe3Sn2 exemplifies an easy-plane system of the class introduced above, in which the spins experience weak anisotropy resulting from the discrete 3-fold rotational symmetry of the rhombohedrally stacked Kagome lattice (12–14). In this work, we study the propagation of spin wavepackets in Fe3Sn2, 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 Fe3Sn2 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 of our study, we use both analytical calculations and numerical modeling to show that these effects are accounted for 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 Fe3Sn2, it applies to antiferromagnetic order as well, for example, the antiferromagnetic version of the theory (20), a quantitative explanation for the recently discovered surprising properties of spin wavepacket propagation in the 2D van der Waals antiferromagnet CrSBr (21).

Prior to measurements of spin transport, the anisotropy parameters of Fe3Sn2 were determined using the time-resolved magneto-optic Kerr effect (TR-MOKE). In this method, spin waves are generated and detected by pulses of light. The amplitude of photoexcited spin waves was found to be independent of polarization, indicating that they are generated by an incoherent mechanism, in which the pump pulse generates a nonequilibrium electron distribution and partial demagnetization. Demagnetization, in turn, leads to a change in the direction of the effective anisotropy field and therefore misalignment between the magnetization, M, and the effective anisotropy field, Heff. The resulting torque causes M to precess, as illustrated in Fig. 1A. The precession leads to oscillations of the component of magnetization parallel to the optic axis, Mz, which are detected via the polar Kerr effect (15, 16, 22). Fig. 1 B and C show oscillations of Mz as detected by the TR-MOKE for several magnetic fields applied in the z direction. Fig. 1D 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 the field is characteristic of a ferromagnet whose biaxial anisotropy can be described by the free energy , where Kx and Kz are the in- and out-of-plane anisotropy energies (Kx, Kz > 0) and x is a preferred magnetization direction within the plane. The origin of the in-plane anisotropy is discussed in SI Appendix, sections I and II, which present a microscopic model whose low-temperature, broken-symmetry phase is described by FA for small fluctuations of the ferromagnetic order parameter. The theoretically predicted dependence of SW frequency on magnetic field, ν(Hz), for a biaxial ferromagnet is (23, 24) where Hs is the saturation field along the z direction, Ms 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 (for a derivation, SI Appendix, section III). The fit (dashed line in Fig. 1D) yields anisotropy parameters Kx ≈ 1.76 × 104 J/m3 and Kz ≈ 2.26 × 105 J/m3, consistent with weak anisotropy within an easy plane, and γ = 1.7 × 1011T−1s−1. The saturation magnetization value, Ms = 6.5 × 105J/m3T−1, was determined from magnetization vs. field measurements (13). Note that while the theoretical prediction (Eq. 1) is written in Gaussian (cgs) units, we have followed the convention of expressing magnetic properties in the SI system.

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. 2A. 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 (25). 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 submicroradian polarization sensitivity. Fig. 2B–I show TR-MOKE maps measured at various pump–probe time delays (Δt) ranging from −7 to +220 ps. Here, the x and y axes refer to the separation between the pump and probe beams. Shortly after photoexcitation (Δt < 100 ps), the transient changes in magnetization are isotropic (Fig. 2C–E). However, at Δt = 180 ps, (Fig. 2F) clear evidence of anisotropic propagation is observed, with a stronger MOKE signal along ∼ < SPSDOUBLEDOLLAR > 20 ° /200° with respect to the x axis. At longer times (Fig. 2G –I), the contrasting nature of propagation between the 20 ° /200° and 110 ° /290° becomes increasingly clear (angular dependence of the anisotropic propagation is further discussed in SI Appendix, section IV). To further characterize spin propagation, we consider the rate of decay of the TR-MOKE oscillations with increasing propagation distance. In Fig. 3A, we plot TR-MOKE time traces for several values of pump–probe separation along the major propagation axis (20 ° /200 °). The amplitude at a given separation is determined from the peak value of the Fourier transform of the oscillations. The log of this amplitude is plotted as a function of (Δx)2 as solid circles in Fig. 3B. For small separations, the amplitude decreases in proportion to e−(Δx/σ)2, where σ ∼ 6 μm. 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 Δx, spin propagation becomes evident; for Δx > 10 μm, the TR-MOKE amplitude deviates from a Gaussian and at Δx = 20 μm 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. Fig. 3C shows the normalized signals for Δx < 10 μm, which is in the Gaussian regime of Fig. 3B. 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 10 μm, shown in Fig. 3D, 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 22 μm reveals another surprising feature. Notice that the first clear indication that spin waves have reached this distance occurs at ≈100 ps, from which we estimate an effective velocity of ≈2 ×107 cm/s. This velocity is six orders of magnitude larger than the group velocity inferred from neutron scattering measurements (26). In the following section, we show that discrepancy is resolved by considering wavepacket propagation in the magnetostatic regime.

As mentioned in the introduction, spin wavepacket propagation in Fe3Sn2 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 that 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 features observed in our spin transport measurements.

We consider a geometry with the equilibrium magnetization in the plane and parallel to one of the easy axes (27, 28). Maxwell’s equations in the magnetostatic regime, ∇ ⋅ B = ∇ × H = 0, together with the Landau–Lifshitz equation, where is the sum of the anisotropy field and the dynamical field , form a closed set that yields the normal modes of magnetization in the long-wavelength regime. To illustrate the resulting MSW dispersion, Fig. 4A shows the calculated spin wave frequency in the kx, ky plane for fixed kz=1 μm−1. Line cuts through this plane defined by kx = 0 (purple) and ky = 0 (orange) plotted in Fig. 4B show forward propagation along the y direction and backward along x, with a saddle point at q = 0. When Kz > Kx, as in Fe3Sn2, the velocity is larger along ky. 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 (SI Appendix, sections V and 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 Fig. 4 C and D as color plots in the time-separation plane. The slope of the lines of the constant phase distinguishes forward- vs. backward-propagating modes. In agreement with our theoretical prediction for the biaxial ferromagnet, modes with wavevector perpendicular to are forward-propagating and backward-propagating for wavevectors parallel to .

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 propagates in the MSW regime. Fig. 5A–C show the comparison of experiment and theory for the amplitude and position of the wavepacket. Fig. 5A presents an expanded view of normalized wavepackets measured at several separations larger than 10 μm; arrows indicate the time at which the peak amplitude reaches a given distance from the pump. The solid circles in Fig. 5B show the displacement of the wavepacket peak as a function of time. Conventionally, the slope of a fit to these points yields the group velocity, vg. From this perspective, the data are quite puzzling, as vg appears to increase with time, reaching ≈2 ×107 cm/s, a value that is much larger than that expected for spin waves. Finally, Fig. 5C 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 Fe3Sn2. Crucially for the interpretation of our experiments, photoexcitation launches a coherent spin wavepacket, comprising 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: with, 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, δp, of the perturbation that induces the subsequent precessional motion. Parameter values, δp = 230 nm and α = 2.7 × 109 s−1, were chosen to achieve the best fit (solid red line) to the amplitude vs. distance data shown in Fig. 5C. The same parameters accurately reproduce the anomalous wavepacket position vs. time data as well (red line in Fig. 5B), adding additional support for our theoretical model (SI Appendix, section V for details).

In the previous section, we showed that spin wavepacket dynamics in Fe3Sn2 can be quantitatively modeled by the MSW dispersion relations for a biaxial ferromagnet. In this section, we offer a physical picture that underlies the most puzzling feature of the wavepacket propagation: an apparent velocity that exceeds 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. 5D. In this approximation to the actual relation (Fig. 4B), SWs propagate with constant velocity for ky < 2kz and do not propagate for ky > 2kz. The precursor effects arise from SW modes in which kzσ is small, such that the Gaussian distribution of photoexcited wavevectors (green line in Fig. 5D) spans both propagating and nonpropagating regimes. The time-evolution of the wavepacket is given by summing the contributions from the two regimes, where ω0 is the frequency at ky = 0 and v is the slope of the V-shaped region. Fig. 5E shows the propagating and nonpropagating terms in Eq. 5 evaluated at t = 0 (red and blue lines, respectively), together with their sum (green line). The individual terms in Eq. 5 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 the 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.

We have shown that spin waves in Fe3Sn2 can be optically excited, propagated, and detected across large distances (> 20 μm) within short timescales (< 100 ps). The observation of precursors reflects a unique regime of light–matter interaction, resulting from the combination of Gaussian laser excitation and V-shaped magnetostatic spin wave dispersion. We note that this phenomenon is distinct from the Sommerfeld-Brillouin precursor (29) or effects associated with propagation in a regime of strong absorption, as observed for spin propagation on lengths scales ≤50 nm in NiO (30). Here, the potential for applications is the ability to transmit a large bandwidth of spin information across macroscopic distances, on a time scale not limited by the group velocity of the spin waves.

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.

The time-resolved magneto-optic Kerr effect (tr-MOKE) measurements with an out-of-plane magnetic field were performed with 1,560-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 0.25. 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 photoelastic modulator (PEM).

The nonlocal 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 surface with approximate spot sizes of 6 μm and 5 μm, respectively, with incident laser powers fixed at 30 μW. This power optimized signal-to-noise ratio without generating measurable light-induced average heating. (SI Appendix, section VII for further information on the pump power dependence of the wavepacket dynamics.) The position of the pump focus was scanned by adjusting the voltage applied to the 2-axis galvanometer-driven 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.

Click here for additional data file.