Inertia-driven resonant excitation of a magnetic skyrmion

Topological spin structures such as magnetic domain walls, vortices, and skyrmions, have been receiving great interest because of their high potential application in various spintronic devices. To utilize them in the future spintronic devices, it is first necessary to understand the dynamics of the topological spin structures. Since inertial effect plays a crucial role in the dynamics of a particle, understanding the inertial effect of topological spin structures is an important task. Here, we report that a strong inertial effect appears steadily when a skyrmion is driven by an oscillating spin-Hall-spin-torque (SHST). We find that the skyrmion exhibits an inertia-driven hypocycloid-type trajectory when it is excited by the oscillating SHST. This motion has not been achieved by an oscillating magnetic field, which only excites the breathing mode without the inertial effect. The distinct inertial effect can be explained in terms of a spin wave excitation in the skyrmion boundary which is induced by the non-uniform SHST. Furthermore, the inertia-driven resonant excitation provides a way of experimentally estimating the inertial mass of the skyrmion. Our results therefore pave the way for the development of skyrmion-based device applications.


Results
Model. To explore the inertia-driven resonant excitation of the skyrmion, we model an interface-induced skyrmion in a ferromagnet (FM)/nonmagnet (NM) bi-layer nanodisk with a diameter of 80 nm. In the simulation, the skyrmion has a Neel-type DW at its boundary due to the interfacial DMI, which inherently exists at the interface between the FM and the NM layers (see Fig. 1a). We excite the skyrmion by applying an oscillating field or current of various frequencies. The excitation mode of the skyrmion can be characterised by the skyrmion position R = (X, Y) from the disk centre, = = + R X Y R 2 2 , and the area of the skyrmion S, as shown in Fig. 1b. In this definition, the increase in the maximum displacement R max and the area variation ΔS in the frequency spectrum correspond to the gyrotropic mode excitation and the breathing mode excitation, respectively.
Breathing mode of a skyrmion induced by an oscillating out-of-plane magnetic field. We first examine the magnetic-field-driven resonant excitation of the skyrmion. To this end, we first locate the skyrmion at the disk centre and then apply an oscillating out-of-plane magnetic field, H f t H e sin (2 ) Figure 1c shows the frequency spectrum of the R max (blue) and ΔS (red), which are normalized by H z . While no significant variation is observed in the R max spectrum, a clear peak is observed in the ΔS spectrum at ≈ . f 5 7 H ac GHz, indicating that the breathing mode is excited at this frequency. Physically, the resonant excitation of the breathing mode originates from the resonant oscillation of the DW at the circular edge of the skyrmion. Therefore, ≈ . f 5 7 H ac GHz corresponds to the resonance frequency of the DW, which is determined by the DW anisotropy energy potential, i.e., the energy difference between Neel-and Bloch-type DW's 27 . It should be noted that there is no inertia-induced hypocycloid-type motion in this resonant breathing mode, indicating that the oscillating magnetic field does not deform the boundary spin structures of the skyrmion. That is to say, the phase of the DW oscillation exactly coincides with the resonant skyrmion breathing motion, so that there is no deformation of the spin structure inside the DW.
Three skyrmion-mode excitations under oscillating SHST. We next investigate the resonant excitation of the skyrmion driven by the spin Hall effect (SHE). We initially locate the skyrmion at the disk centre and then apply an in-plane AC current of frequency f ac along the NM layer. Then the spins accumulated at the interface between the FM and the NM due to the SHE of the NM layer diffuse into the FM layer and exert an oscillating SHST to the skyrmion in the FM layer (see Methods). Figure 2 shows the frequency spectra of the skyrmion-mode excitations under the oscillating SHST. As we increase the f ac , three different excitation modes are observed. At a relatively low frequency regime, a sharp peak is observed in R max at f ac ≈ 0.26 GHz (regime I: the area shown in red). With a further increase in the f ac , we observe a sharp peak in ΔS at f ac ≈ 2.8 GHz (regime II: the area shown in blue), and a broad peak both in R max and ΔS at f ac = 5 ~ 6 GHz (regime III: the area shown in yellow).
Regime I: gyrotropic mode. In order to understand the observed frequency spectra, the trajectories of the skyrmion and corresponding snapshot images are extracted from the simulation and plotted in Fig. 3 (see also Supplementary Movies). It is clear that the resonant peak in R max at f ac ≈ 0.26 GHz corresponds to the gyrotropic motion (Fig. 3a), which is typically seen in two dimensional topological spin structures, such as vortices. This gyrotropic motion can be described by Thiele's equation 28 as where = G G (0, 0, ) is the gyrocoupling vector, k is the effective spring constant induced by a geometrical potential, and D is the dissipation tensor, and  R is the time derivative of the position R. The gyrotropic motion of the skyrmion is analogous to a massless particle with an electrical charge in a magnetic field and an external potential through a viscous medium 15 . If we neglect the dissipation term, the resonance frequency, f c , can be easily calculated to be π = f k G 2 / c . Therefore, the resonant gyrotropic motion can be observed at = f f ac c . This corresponds to the excitation mode observed in regime I. It should be noted that there is no inertial effect in regime I because the gyrotropic motion does not accompany the deformation of spin structures at the skyrmion boundary.
Regime II: breathing mode. In regime II, on the other hand, a clear peak is observed in ΔS at f ac ≈ 2.8 GHz, indicating the resonant excitation of the breathing mode, which is demonstrated in Fig. 3b. This breathing mode is similar to the field-driven resonant excitation as shown in Fig. 1c. However, the resonance frequency is exactly half of that for the field-driven case ( f H ac ≈ 5.7 GHz). Such a difference in resonance frequency originates from the confinement effect of the skyrmion in a circular disk. Unlike a magnetic field, the SHST generally induces a transverse motion in the skyrmion (which is known as the skyrmion Hall effect) 29 . As a result, the skyrmion centre moves along the transverse direction, generating an elliptical trajectory (see the trajectory in Fig. 3b). When the skyrmion deviates from the centre of the disk, it experiences a geometry-induced magnetic field whose frequency is twice that of the f ac due to the elliptical trajectory. This is the reason why we observe a resonant breathing mode GHz. This implies that the resonant breathing mode observed at ≈ . f 2 8 ac GHz can be understood as a resonant excitation by the geometry-induced field. Like the gyrotropic mode (regime I), the inertial effect does not appear in the resonant breathing mode.
Regime III: inertia-driven hypocycloid-type mode. Contrary to regimes I and II, a broad peak is observed both in R max and ΔS at f 5 6 ac = ∼ GHz, which implies that the breathing mode and the gyrotropic mode are simultaneously excited. The corresponding skyrmion trajectory and snapshots are shown in Fig. 3c. This result is clearly distinguished from the magnetic-field-driven case in a similar frequency regime, where only the breathing mode is excited (Fig. 1c). The clear hypocycloid-type trajectory observed in regime III indicates that the inertia-driven resonant excitation of the skyrmion is achieved by the oscillating SHST.
According to a recent theory 2 , the inertial effect of a skyrmion can be described by inserting the Newtonian mass (M) term into Thiele's equation as If we neglect the dissipation term (D = 0), Eq. (2) gives two circular modes with frequencies, 2 In the first order approximation with a small M, each frequency can be reduced to This indicates that the inertial mass simultaneously creates two circular modes with an opposite sign of frequency. One mode gives a counterclockwise (CCW) rotation with a relatively low frequency (ω + ), and the other mode gives a clockwise (CW) rotation with a relatively high frequency (ω − ). The combination of the two circular motions generates a hypocycloidal motion, which we observed in regime III. Note that the low-lying mode (ω + ) is identical to the resonant gyrotropic mode which we observed in regime I.
Fundamentally, the two circular modes induced by inertial mass can be understood in terms of the additional spin degree of freedom at the skyrmion boundary. If we look at the spin structure at the circular boundary of the skyrmion (i.e., the boundary DW), two circular modes correspond to the excited spin waves which propagate in the opposite directions inside the DW channel 2 . One is a slow spin wave propagating along the CCW direction, and the other is a fast spin wave propagating along the CW direction. For the slow spin wave, the spins adiabatically align with the direction of the DW, thus it is insensitive to the inertia (corresponding to the low-lying gyrotropic mode).
For the fast spin wave, on the other hand, the spin oscillation is out of phase with the direction of the DW, therefore a strong inertial effect emerges (see Fig. 3d). This implies that the spin wave excitation in the DW channel is necessary to observe the inertial effect. This consideration allows us to understand why the strong inertial effect is only observed with the oscillating SHST, and not the oscillating magnetic field. The magnetic field gives a global torque to the boundary DW, so that the entire spin exhibits a coherent motion, which leads to a breathing motion. The SHST-induced torque, on the other hand, depends on the magnetization direction, so that it exerts a non-uniform torque to the boundary magnetization, which can trigger the spin wave excitation in the DW channel. We note that an unusual drop is observed at the end of region III, as shown in Fig. 2a,b. We think that this is due to the confined geometry effect since the R max and ΔS increase rapidly and the disk size starts to affect the skyrmion motion.

Discussion
The inertia-driven resonant excitation of the skyrmion in regime III provides a way of estimating the inertial mass. According to Eq. (3), the inertial mass can be calculated from the sum of the frequencies of the two circular modes: M = − G/ (ω + + ω − ). In Fig. 4, we summarize the inertial mass and mass density as a function of the frequency f ac . Up to this point, the inertial effect has only been observed in the relaxation process. Thus, it is not easy to extract the inertial mass because one needs to extract both frequencies during the short relaxation process. However, our results provide a more convenient way to determine the inertial mass because one of the frequencies (the higher frequency) is fixed by the external oscillating source. Furthermore, the inertial effect appears constant during the resonant excitation, which is greatly beneficial to the experimental approaches. We note that to consider more realistic situations, further study will need to take into account the thermal-fluctuation effects. However, the key finding in our work would still hold even at finite temperature, because the zero temperature model can well describe the inertia-driven skyrmion motion at finite temperature 2,24 .
In conclusion, we found that the inertial effect of a skyrmion can be observed by applying an oscillating SHST. The intriguing observation can be attributed to the additional degree of freedom at the boundary of the skyrmion. Because the SHST exerts a non-uniform torque on the boundary spin of the skyrmion, the spin waves are excited and propagate along the boundary, which forms an inertia-driven hypocycloidal trajectory. Our result also provides a convenient way to estimate the inertial mass of the skyrmion, which is crucial to understanding the motion of the skyrmion. This work therefore paves the way for the development of skyrmion-based spintronic devices.

Methods
The micromagnetic simulations were performed using the OOMMF code 30 with the DMI extension 31 . In a continuum model, the interfacial DMI energy density can be expressed as 7,21 D x y e m m e m m , We adopted a model system of a thin FM disk with a diameter of 80 nm on a HM electrode; e.g., Pt(HM)/ Co(FM), which induces broken inversion symmetry in the FM disk through the spin-orbit interaction with the bottom HM layer (see Fig. 1a). To realize steady motion of the skyrmion through excitation, an alternating current (AC) at various frequencies was applied along the HM electrode layer to induce an oscillating SHE, which exerts a SHST to the FM layer.
The governing equation of the simulation is the Landau-Lifshitz-Gilbert equation with the time-varying SHST term. It is described as 32  where γ is the gyromagnetic ratio; f ac is the AC frequency; and H eff is the effective magnetic field stemming from the exchange, the anisotropy, the demagnetization and the DMI energy terms. The third term on the right side of equation (6) is the time-varying SHST term given the assumption that the AC is applied along the in-plane x direction. The coefficient H SH is a time-independent parameter which is defined as 32,33 H j eM t , z SH B SH c s μ θ γ = where θ SH is the spin Hall angle of the adjacent HM layer, μ B is the Bohr magneton, e is the electron charge, t z is the thickness of the FM layer (t z = 0.6 nm in our simulations), and j c is the AC amplitude which flows in the x direction. It was also assumed that most electrons flowed through the HM under layer (with a thickness typically of 3~5 nm), as the thickness and the conductivity of the HM layer can be much greater than those of the FM layer 33 . In this study, therefore, we only considered the SHST and neglected spin-transfer torques as well as the Rashba effect 32,33 . Throughout this study, the amplitude of the SHST was set to H SH = 0.1 × 10 4 A/m for the MHz-frequency and H SH = 2.69 × 10 4 A/m for the GHz-frequency excitations. In our nanodisk system, the skyrmion has a Neel-type domain wall (a hedgehog-type skyrmion), and the skyrmion topological number is . The area, S, and the position of the skyrmion, (X, Y), were defined by the area of the region where the out-of-plane magnetization component m z is greater than 0 and by the centre of the area, respectively (see Supplementary note 1). Note that the radial position of the skyrmion is R = (X 2 + Y 2 ) 1/2 (see Fig. 1b). The frequency spectra were obtained using the fast Fourier transform. See Supplementary note 2 for detailed information about the simulation method.