Giant effective damping of octupole oscillation in an antiferromagnetic Weyl semimetal

A magnetic Weyl semimetal is a recent focus of extensive research as it may exhibit large and robust transport phenomena associated with topologically protected Weyl points in momentum space. Since a magnetic texture provides a handle for the configuration of the Weyl points and its transport response, understanding of magnetic dynamics should form a basis of future control of a topological magnet. Mn3Sn is an example of an antiferromagnetic Weyl semimetal that exhibits a large response comparable to the one observed in ferromagnets despite a vanishingly small magnetization. The non-collinear spin order in Mn3Sn can be viewed as a ferroic order of cluster magnetic octupole and breaks the time-reversal symmetry, stabilizing Weyl points and the significantly enhanced Berry curvature near the Fermi energy. Here we report our first observation of time-resolved octupole oscillation in Mn3Sn. In particular, we find the giant effective damping of the octupole dynamics, and it is feasible to conduct an ultrafast switching at<10 ps, a hundred times faster than the case of spin-magnetization in a ferromagnet. Moreover, high domain wall velocity over 10 km/s is theoretically predicted. Our work paves the path towards realizing ultrafast electronic devices using the topological antiferromagnet.

A magnetic Weyl semimetal is a recent focus of extensive research as it may exhibit large and robust transport phenomena associated with topologically protected Weyl points in momentum space. Since a magnetic texture provides a handle for the configuration of the Weyl points and its transport response, understanding of magnetic dynamics should form a basis of future control of a topological magnet. Mn3Sn is an example of an antiferromagnetic Weyl semimetal that exhibits a large response comparable to the one observed in ferromagnets despite a vanishingly small magnetization. The non-collinear spin order in Mn3Sn can be viewed as a ferroic order of cluster magnetic octupole and breaks the time-reversal symmetry, stabilizing Weyl points and the significantly enhanced Berry curvature near the Fermi energy. Here we report our first observation of time-resolved octupole oscillation in Mn3Sn. In particular, we find the giant effective damping of the octupole dynamics, and it is feasible to conduct an ultrafast switching at < 10 ps, a hundred times faster than the case of spin-magnetization in a ferromagnet.
Moreover, high domain wall velocity over 10 km/s is theoretically predicted. Our work paves the path towards realizing ultrafast electronic devices using the topological antiferromagnet.
In the field of spintronics, antiferromagnetic (AF) metals have attracted significant attention as next-generation active materials of electronic devices for their vanishingly small stray field perturbing neighboring cells. The recent rapid development in AF spintronics [19][20][21] has led to the demonstration of electric reading and writing of an AF state [22,23], which has been recently further supported by several kinds of AF domain imaging techniques [24][25][26][27][28]. Because of its vanishingly small magnetization, the detection means for such an AF metallic state has been restricted to anisotropic magnetoconductance [22,23], quadratic magneto-optical effects [29], and resonant X-ray diffraction [30], which are far weaker than the magnetization M-linear response such as AHE and magneto-optical effects employed for ferromagnets.
With this respect, recently discovered D019-Mn3Sn stands out as a unique antiferromagnet that exhibits large electric and optical M-linear responses such as AHE [10], ANE [13,14], and magneto-optical Kerr effect (MOKE) [31] even though it has only a vanishingly small magnetization. Significantly, it has been clarified that Mn3Sn is a magnetic Weyl semimetal [10,13,15]. The non-collinear chiral magnetic texture in Mn3Sn can be viewed as a ferroic order of a cluster magnetic octupole and breaks TRS macroscopically [32]. Figure 1a shows the crystal and magnetic structures of Mn3Sn.
The magnetic moments of Mn lie in the (0001) plane and form an inverse triangular spin structure. In this structure, one unit of the cluster magnetic octupole is made of the six neighboring moments on an octahedron (bi-layered triangle) of Mn atoms as featured by a colored hexagon in Figs. 1a and 1b. By 180̊ rotation of each spin, the octupole polarization may reverse its direction (Fig. 1b). As a result, the in-plane non-collinear spin order can be viewed as Q = 0 order of the magnetic octupole. The AF spin texture with this ferroic order induces large Berry curvature due to Weyl points in momentum space [13,15], leading to the large transverse response. In this regard, the observation of the spin dynamics in a topological magnet would be a key step for future manipulation of the Weyl points in the momentum space and the associated large responses. In addition, similarly to the case of the spin-magnetization dynamics in ferromagnetic metals, understanding and control of the time-dependence of the cluster magnetic octupole in a chiral AF metal would be an important step for developing the device physics in magnetism. To date, there has been no report on the time-resolved observation of spin dynamics in either ferromagnetic or antiferromagnetic Weyl semimetals. Even when we focus on the previous research on antiferromagnets [29,30,[33][34][35][36][37], the direct, time-resolved observation of the spin precession has been limited to AF insulators [35-37] and never been made in AF metals. A few papers report the time-resolved dynamics of the order parameter in AF metals [29,30,34] but never been able to associate them with the spin-wave modes. In this article, we report our observation of the time-resolved spin dynamics in the AF Weyl semimetal Mn3Sn.
Let us first discuss the case of collective spins in ferromagnetic materials, where each spin is coupled in parallel by exchange interaction (Supporting Information 1.). A precessional motion of each spin is always in-phase as schematically shown in Fig. 1c, and thus the energy scale of the resonant frequency (ħω ~ K) is independent of the exchange interaction and is determined by K only, where K is magnetic anisotropy energy mainly originating from on-site spin-orbit interaction. A typical time scale for the magnetization switching, expressed as (Δω) −1 , can be ~(2αK) −1 ħ. Here Δω and α are spectral linewidth and effective damping constant, respectively. Generally, the time scale is longer than 1 ns (e.g. ω = 10 GHz and α = 0.1).
To consider collective spins in a chiral AF metal (Supporting Information 1.), the following Hamiltonian to treat the inverse triangular spin structure can be employed Here, S, J and D denote spin-angular momentum, exchange interaction and Dzyaloshinskii-Moriya interaction, respectively. (i, j) and (a, b) refer to the Mn sites and one of the three sublattices (A, B, and C) of the inverse triangular lattice structure, respectively. K is introduced to describe six-fold magnetic anisotropy in Mn3Sn with ka = (cosψa, sinψa, 0) and (ψA, ψB, ψC) = (0, 4π/3, 2π/3). εab is the antisymmetric tensor which satisfies εAB = εBC = εCA=1 and z is the unit vector along the c-axis. Note that in-plane and out-of-plane magnetic anisotropies from the kagome plane should be determined by K and D, respectively. Figures 1d and 1e   [41], and can be reversed by an external magnetic field. Note that the appearance of the anomalous Hall effect [10], anomalous Nernst effect [13,14], and MOKE [31] is not induced by the spin-magnetization due to the canting but by the cluster magnetic octupole. As shown in Fig. 2a where ħ, α, and alat indicate the reduced Planck constant, damping constant, and the lattice constant of the nearest neighbor Mn atoms, respectively. Here S is the size of spin angular momentum ~1.5 for the Mn magnetic moment ~3μB, ϕ refers to in-plane precession angle of Mn magnetic moment, and α is identical to the Gilbert damping constant in the Landau-Lifshitz-Gilbert equation. From Eq. 2, resonant frequencies for modes I and II can be estimated as follows: Here the resonant frequencies for the modes I and II (ω I and ω II ) correspond to the optical and collective precession-like modes depicted in Figs. 1d and 1e, respectively.
The damped oscillation is often expressed by exp(−α I(II) ω I(II) t) by introducing a phenomenological effective damping constant (αI, αII), which is expressed as Note that the effective damping (αI, αII) is not identical to the Gilbert damping constant However, the resonant frequency of the collective precession-like mode (ωII) is not.
This is because the cluster magnetic octupole can couple with an external magnetic field via spontaneous magnetization due to canting. From the field dependence of the resonant frequencies (Fig. 3d), ωII/2π at B = 0 is determined to be 13.7±1.5 GHz.
To characterize the dynamics of the cluster magnetic octupole, the six-fold magnetic anisotropy K was determined from the torque measurements (Supporting Information 7.). Figure 4a shows  As discussed, the dynamics characteristic of the cluster magnetic octupoles is an ultrafast damped oscillation due to the exchange-interaction (Eq. 5). Interestingly, the exchange-interaction dramatically increases the effective damping (αII = 1.0). Thus, the typical switching time can be ultrafast (1/ωIαI ≈ 1/ωIIαII ~ 9 ps) although the resonant frequency for the collective precession-like motion is relatively slow (ωII/2π = 13 GHz).

Resonant frequency and spectral linewidth
When we treat the spherical coordinate system (θ, ϕ) to describe spin-direction, resonant frequency (ω) and spectral linewidth (Δω) can be derived as follows from (S4) Here, U, α, and S are potential energy, Gilbert damping constant and spin angular momentum, respectively. because in-phase ϕ-motion and out-of-phase θ-motion are determined by K and J, respectively.

Sample preparation
Polycrystalline samples were prepared by melting the mixtures of Mn and Sn in an Al2O3 crucible sealed in an evacuated quartz ampoule in a box furnace at 1050 ˚C for 6 h. In preparation for single-crystal growth, the obtained polycrystalline materials were crushed into powders, compacted into pellets, and inserted into an Al2O3 crucible that was subsequently sealed in an evacuated SiO2 ampoule. Single-crystal growth was performed using a single-zone Bridgman furnace with a maximum temperature of 1080 ˚C and growth speed of 1.5 mm h −1 . These are exactly the same methods for fabricating the single crystals as those used for the previous study on the Kerr effect [2] and the chiral anomaly due to Weyl fermions [3]. Analysis using inductively coupled plasma spectroscopy showed that the composition of the single crystal is Mn3.07Sn0.93. The bulk-Mn3Sn sample was cut and polished so that the sample had optically smooth surfaces along the   2 1 10 plane. Then the sample was annealed at 600 ˚C under vacuum (~2×10 −6 Pa) for 1 h. Without breaking vacuum, 4-nm-AlOx was subsequently prepared onto the Mn3Sn surface by electron-beam deposition method to prevent degradation during MOKE measurements.

Time-resolved magneto-optical Kerr effect (TR-MOKE) measurements
Time-resolved polar MOKE signals were measured in a conventional all-optical pump-probe setup using a Ti-sapphire laser with a regenerated amplifier [4]. grain is randomly oriented in the film, coherent excitation should not be made. Here, the TR-MOKE experiment is done with the setup and conditions that are different from those employed in the main text using bulk-Mn3Sn. The laser wavelength, pulse width, and repetition rate are 800 nm, 140 fs, 80 MHz, respectively. Pump fluence is ~10 times smaller than that employed in Fig. 3 in the main text. Because of the relatively small Kerr rotation angle for the 800 nm-wavelength (Fig. S1 inset), we have employed a different pulse laser system from the one employed in the main text. We employed magnetron sputtering to prepare a polycrystalline Mn3Sn thin film and confirmed that the film exhibits anomalous Hall effect and MOKE signal due to the cluster magnetic octupole (Fig. S1 inset). The sample fabrication procedure can be found elsewhere [5]. As shown in Fig. S1, TR-MOKE results indicate the absence of the oscillation signals attributed to mode II. The black solid curve shows a fit using an exponential function. The same background is used for the one in Fig. 3c in the main text.
In Fig. 3c of the main text, there is an oscillation-like signal around 10 ps, which is apparently different from the oscillation signals originating from the modes I and II. As explained above, TR-MOKE results in Fig. S1 indicate the absence of the oscillation signals attributed to modes I and II but a similar oscillation-like signal around 10 ps remains. As this point, we do not have enough information to identify its origin. However, it should be noted that the signal has nothing to do with the coherent spin-wave oscillation.

Coherent spin-wave excitation in Mn3Sn
During the TR-MOKE measurements, an external static magnetic field is applied to direct the octupole polarization along the magnetic field direction. Moreover, the static magnetic field is necessary to adjust the initial phase of each Mn spin to an identical one, which is indispensable for observation of mode I. Figure S2a shows a ground state of the spin structure of Mn3Sn under zero-field while Fig. S2b presents the configuration stabilized under an external magnetic field. As shown in Fig. S2b, an external magnetic field induces the canting of the magnetic moments and aligns polarization of all the octupoles along with the same direction as the initial phase depicted in Fig. 1d in the main text. When the parameters (J, D, and K) are decreased by a pump pulse, the canting angle increases, and the precession can be driven coherently. interactions. Therefore, the pump pulse increases a spin-canting induced by an external magnetic field (B) as explained above. In other words, the pump pulse creates an effective magnetic field parallel to B, following the similar mechanism to the case of a ferromagnet [6].
As a result, the torque to drive a coherent precession arises along out-of-kagome-plane direction as indicated in red (Fig. S3a, see below). When the coherent precession starts, the octupole order parameter reduces. For the case of Fig. 3b in the main text, the phase delay (ϕ0) cannot be neglected because the transition time for the recovery (τ ~ 0.7 ps) is comparable to the oscillation period (~1.2 ps (ωI/2π=0.86 THz)). Here, let us assume the exponential decay of the torque on each magnetic moment (as shown in Fig. S3b) because this is the simplest and most natural temporal profile of the torque. Then the phase delay can be derived to be ϕ0 = tan −1 (ωIτ) as depicted in Fig. S3c [7]. For the observed recovery time of τ = 0.7 ps, a phase delay is estimated to be around 70-80 deg., which is close to the maximum one (ϕ0 = 90 deg.).
Notably, after considering the phase delay, the simulated Kerr rotation angle (solid curve in Fig. S3a) well reproduces the observed oscillation signal in Fig. 3b in the main text.

Theoretical study to determine the spin-wave modes
Spin-wave excitations in Eq. 1 were calculated on the inverse triangular spin structure.
First of all, we transform the local frame such that its z-axis is parallel to the classical spin direction of the ground state [8]. and aia = (aia, a † ia, S − a † ia aia). Here, the annihilation (creation) operator aia (a † ia) represents the Holstein-Primakoff boson, and the Hamiltonian (Eq. 1 in the main text) is expressed in terms of aia and a † ia. Within the linear spin-wave approximation, we neglect higher-order terms of aia and a † ia, and calculate spin-wave excitation energies keeping them up to the quadratic order. Figure S4 shows the schematic diagram of the detailed torque measurements setup of the in-plane magnetic field rotation in the kagome-lattice of Mn3Sn. The setup corresponds to the experiment for Fig. 4b  where U represents magnetic anisotropy energy. Here, K2/2 (= 3. and six-fold magnetic anisotropy. Therese values are comparable to those obtained in the previous work [9].

Analysis for the torque measurements
From the crystal structure of D019-Mn3Sn, only six-fold magnetic anisotropy should appear. However, we find that in addition to the K 6 term, K 2 and K 4 terms are indispensable to reproduce the experimentally obtained torque data and are strongly field-dependent. Therefore, K2 and K4 terms would be attributed to magnetostriction in Mn3Sn, and an intrinsic magnetic anisotropy at B = 0 should be six-fold only, being described by K6/18 = 3.
For the case of Mn 3 Sn, the domain wall propagates with a high velocity without showing the Walker breakdown because of strong magnetic anisotropy due to the Dzyaloshinskii-Moriya interaction, which prevents the out-of-kagome-plane motion (Fig. 4a in the main text) [10].