Elongated Skyrmion as Spin Torque Nano-Oscillator and Magnonic Waveguide

Spin torque nano-oscillator has been extensively studied both theoretically and experimentally in recent decades due to its potential applications in future microwave communication technology and neuromorphic computing. In this work, we present a skyrmion-based spin torque nano-oscillator driven by a spatially uniform direct current, where the skyrmion is confined by two pinning sites. Different from other skyrmion-based oscillators that arise from the circular motion or the breathing mode of a skyrmion, the steady-state oscillatory motions are produced by the periodic deformation of an elongated skyrmion. Through micromagnetic simulations, we find that the oscillation frequency depends on the driving current, the damping constant as well as the characteristics of pinning sites. This nonlinear response to direct current turns out to be universal and can also appear in the case of antiskyrmions, skyrmioniums and domain walls. Furthermore, the elongated skyrmion possesses a rectangle-like domain wall, which could also serve as a magnonic waveguide. Utilizing the propagation of spin waves in this waveguide, we propose a device design of logic gate and demonstrate its performance.


Introduction
as well as the competition between the spin torques and inherent damping yield a sustained oscillation of magnetization. The frequency can reach a few GHz that is higher than most of other ferromagnetic skyrmion-based oscillators (around a few hundred MHz). The oscillation is also caused by the deformation of a skyrmion, but the driving current is spatially uniform, which is easily accessible compared to the method reported in other work [26]. Furthermore, both theoretical and experimental results have shown that a domain wall naturally acts as a magnonic waveguide, where spin waves can be excited easily and propagate with low dissipation [55][56][57][58]. In our model, the elongated skyrmion involves a rectangle-like domain wall so that the spin wave can propagate in this channel. From the application point of view, both STNOs and magnonic waveguides are promising building blocks for the next-generation information processing and storage devices, as well as the pattern recognition and classification in neuromorphic computing [59][60][61][62][63][64].

Results and Discussion
Model and initial state. Here, we consider a ferromagnetic nanotrack grown on a heavy metal that provides sufficiently strong Dzyaloshinskii-Moriya interaction (DMI) [65,66] to stabilize chiral skyrmion-like structures and allows the spin Hall effect when an electric current flows through it due to the strong spin-orbit interaction. To obtain an elongated skyrmion, two artificial pinning sites are placed in the nanotrack, which are constructed by a local perpendicular magnetic anisotropy (PMA) reduction and indicated by the dark gray region of width w in Fig. 1(a). The PMA constant in pinning sites is K p = 0.55 MJ·m −3 and the width is w = 10 nm in the following simulation, unless specified otherwise. Such a modification of PMA considered in this device is due to the fact that the local PMA reduction can generate an energy well to attract or trap skyrmions and act as a pinning site [67][68][69]. Recent theoretical work has also reported the formation of square-shaped skyrmions in a magnetic thin film with orthogonal defect lines formed by the reduction of PMA [70]. In experiments, this type of pinning sites can be created by locally applying additional sputtered layers [71] or by using ion irradiation [72].
We begin with the nucleation of an elongated skyrmion. At the first step, a single spin-polarized current pulse is perpendicularly injected into the central region indicated by the dashed box in Fig. 1(a) to overcoming the topological stability barrier. The current density and the duration of this pulse are 300 MA·cm −2 and 0.2 ns, respectively. Then, the current is switched off, and a stable elongated skyrmion confined in the region between two pinning sites is obtained after a sufficiently long-time relaxation (see Supplementary Note 1 for more details). This current-induced genera- tion of skyrmions is one of the most efficient schemes for achieving the reproducible nucleation of individual skyrmions at a given position [47], which has also been experimentally realized via vertical current injection through a scanning tunneling microscope tip [73], 3D conducting path [74] or magnetic tunnel junctions [75]. The rapid progress in this field provides promising future for the experimental realization of the proposed device. Figure 1(b) shows the spin configuration of a relaxed elongated skyrmion. Indeed, the formation of this non-circular spin texture can be seen as a circle-shaped skyrmion elongated by two forces from the pinning sites since the region with reduced PMA tends to attract a domain wall or a skyrmion as mentioned before. Supplementary Note 2 provides details of the conversion between a circle-shaped ordinary skyrmion and an elongated skyrmion.
Oscillations of a domain wall and an elongated skyrmion. Next, when the driving current is applied to the heavy metal along x axis, the elongated skyrmion will deform periodically and generate a stable spin oscillation that is detected by the magnetoresistance effect through a magnetic tunnel junction. To quantify this dynamical behavior, we plot the temporal evolution of m z as shown in Fig. 1(c), where · · · represents the spatial average of a component of magnetization over the entire nanotrack, and calculate the oscillation frequency by using the fast Fourier transform (FFT). From the frequency spectra in Fig. 1(d), it is noted that some higher-order harmonics are also excited, in addition to the fundamental frequency. Therefore, the profile of m z is not a pure sine-like curve, as illustrated in Fig. 1(c).
To understand the skyrmion oscillation, we first consider the case where a single domain wall is located at the center of a nanotrack as depicted in Fig. 2(a). When the spin current is applied parallel to the domain wall (i.e., the corresponding unit polarization vector is p = ±ŷ, since p should be perpendicular to the current flow in the case of spin Hall effect), the domain wall can not be driven and remain unchanged. However, if there is a perturbation to force the partial domain wall to deform and deviate from the equilibrium at the initial state as shown in Fig. 2(b), the domain wall will create a sustained oscillation driven by the spin current along the x axis, similar to an ordinary rope [see Fig. 2(c) and Supplementary Movie 1]. Here, the oscillation is attributed to the competition between the spin torque, the intrinsic damping and the boundary barrier. Besides, such a perturbation can be realized by locally applying an external field or a driving current along y axis. However, it should be noted that the oscillation occurs only when the spin polarization of the driving current is opposite to the orientation of the y component of magnetization in the domain wall, e.g., p is equal to −ŷ for the case shown in Fig. 2(a). Otherwise, the domain wall will restore to equilibrium and keep steady. As a result, the elongated skyrmion is considered as an oscillation source in this work with two significant advantages: 1) it possesses natural domain walls with curvature so that no artificial initialization is required to produce a perturbation; 2) Whether the spin polarization is +ŷ or −ŷ, the oscillation is detected, since the elongated skyrmion has two types of domain walls, corresponding to the reversal of the magnetization from +ẑ to −ẑ and from −ẑ to +ẑ, respectively.    Fig. 4(b), when the driving current density j is smaller than the first critical value j 1 , the elongated skyrmion is completely pinned, where the initial m z oscillation gradually decays to 0, indicating that the driving force is insufficient to overcome the intrinsic force due to the topological protection and the damping force. However, if the current density is larger than the second critical value j 2 , the elongated skyrmion will depin from the pinning sites and be annihilated around the upper boundary of the nanotrack [see a large current. In the range of j 1 < j < j 2 , the steady skyrmion oscillation occurs, and the frequency increases with increasing j. The dependence of oscillation on the damping coefficient α is also shown in Fig. 4(c). There are still three phases, but, on the contrary, the pinned and annihilation states occur in the high-damping and low-damping areas, respectively, and the frequency is inversely proportional to α in the oscillation region. For a rigid topological object driven by a spin-polarized current, the Thiele equation is given by [77] where the first term represents the Magnus force with G = (0, 0, G) and G = 4πQ, D is the dis-sipative tensor with D i j = δ i j D, F j denotes the current-induced driving force proportional to the current density j and F r is the repulsive force imposed by the boundary. If this driving current is applied along x axis with p = −ŷ and F r = F rŷ , we obtain the following velocity relationship: When the topological body moves along the upper boundary of the nanotrack, the Magnus force is balanced by the repulsive force from the edge, v y = 0 and v x = F j /(αD). It should be noted that the velocity is proportional to the driving current density and increases as the damping constant decreases. From the perspective of qualitative analysis, a fast motion of the marked half-skyrmions in Fig. 3(b) requires a large current density and low damping, which can reduce the motion period and hence increase the oscillation frequency. One can understand why the oscillation frequency increases with increasing j and is inversely proportional to the damping constant.
Here, we continue to investigate the skyrmion oscillation driven by a spin-polarized current as a function of the magnetic parameters. It should be mentioned that the Heisenberg exchange interaction tends to force the magnetic moments of adjacent atoms to align with one another in parallel, while the DMI prefers them to form a non-collinear structure. Therefore, a weak Heisenberg exchange interaction and a strong DMI allow the elongated skyrmion to deform easily and then produce a oscillatory motion of magnetization as shown in Figs. 5(a) and (b). Meanwhile, the total energy versus the exchange stiffness A and the DMI constant D as well as the spin configurations with different magnetic parameters are also given in Supplementary Figure 5, suggesting that the system with a small A or a large D has a lower energy and is more likely to be excited to oscillate. Figures 5(c) and (d) show the effect of the characteristics of pinning sites on the oscillation. It is noted that for each of driving current density j, the increase in K p yields a reduction in the oscillation frequency, and the working window of j is wider in low-PMA region, which is attributed to the strong pinning force caused by the large PMA difference between the pinning site and the clean region. On the other hand, the response of this oscillation to the width of pinning sites is non-monotonic, and there is an optimal size that corresponds to the maximum frequency.
For a narrow pinning region, the strength of the pinning force is not strong enough to protect the skyrmion from annihilation. However, if the width w is too large, most part of the skyrmion will be pinned, thus suppressing its deformation and oscillation. Consequently, to obtain a high frequency of the proposed skyrmion oscillator, the pinning site with low PMA and an appropriate optimal  Excited by a low-frequency microwave stimuli, the magnetic moments in this region are easier to precess than that outside of the domain wall, thus confining spin waves in this narrow channel and forming a waveguide. In addition, considering that the directions of the magnetization or the effective field in the upper and lower branches are different [see Fig. 1(b) and Fig. 7(b)], one can understand why the propagation of excited spin waves is asymmetrical in Fig. 6(b). To further obtain the whole picture of the characteristics of wave propagation, the dispersion relations along two branches are also calculated, as shown in Figs. 7(c) and (d). The propagation of spin waves in such a ring-shaped waveguide will provide alternatives for designing of spintronics devices in future data processing and computing. Taking the logic gate as an example, we propose a device design to demonstrate its performance in Supplementary Note 4, which realizes the basic operations of logic AND, OR and XOR functions.

Conclusions
In this work, we demonstrate the current-induced oscillation of an elongated skyrmion on a nanotrack with two pinning sites. Combining the Thiele equation and the motion of half-skyrmions, the stable oscillation caused by the periodic deformation of an elongated skyrmion is described in detail. We show the effect of the driving current density, the damping coefficient, the magnetic properties and two primary parameters of the pinning region on the performance of the skyrmion oscillator. The working window of this oscillation is also determined. Furthermore, the waveguide properties of such an elongated skyrmion are corroborated through micromagnetic simulations.
Our results reveal an alternative concept of spin-torque oscillator based on a localized magnetic soliton, i.e., the elongated skyrmion. The resulting oscillation exhibits a nonlinear response to the direct current input, which is promising to be used in bio-inspired hardware. The utilization of nanoscale spintronic oscillators for high-performance neuromorphic computing has already been demonstrated in experiments [59][60][61]. For skyrmionics, from the experimental realization perspective, the discovery of ultra-thin magnetic materials with strong DMI for stabilizing isolated skyrmions and the development of nanofabrication techniques, would rapidly accelerate the skyrmion-based device optimization and its applications in future digital information technology. Our work provides guidelines for the development and future experimental exploration of skyrmion-based oscillators.

Methods
Micromagnetic simulations. In this work, we use the public code project object-oriented micromagnetic framework (OOMMF) [80] to study the dynamics of magnetization based on the Landau-Lifshitz-Gilbert equation, The four terms on the right side of the above equation describe the gyromagnetic precession, the dissipation originating from the nonzero Gilbert damping, and the spin torques induced by the driving current, respectively. It should be noted that we focus on the damping-like spin torque τ d = γu d m × (p × m) and ignore the field-like torque τ f = γu f p × m in the main text since the latter is regarded as a weak term in the considered material system without a sufficiently strong interfacial Rashba effect. However, without loss of generality, the effect of the field-like torque on the oscillation has also been investigated and is shown in Supplementary Figure 8. It is found that a large field-like torque can decrease or increase the oscillation frequency, depending on the sign of u f /u d . Here, m denotes the local normalized magnetization, γ is the gyromagnetic ratio, α is the damping constant and p is the spin polarization direction of the spin current. u d and u f are proportional to the current density j, and describe the magnitudes of the damping-like torque and field-like torque, respectively. H eff represents the effective magnetic field associated with the total energy of the system, where A is the Heisenberg exchange stiffness, K is the PMA constant and ε d denotes the demagnetization energy density. The last term arises from the asymmetric exchange interaction DMI, which is related to the spin configuration and the chirality of magnetic structures. For example, the Bloch-type skyrmion is stabilized by the bulk DMI ε DMI = Dm · (∇ × m), the Néel-type skyrmion is formed in the thin film with the interfacial DMI ε DMI = D[(m · ∇)m z − m z (∇ · m)], while for the magnetic system with crystallographic class D 2d that tends to stabilize an antiskyrmion, the DMI energy density is given by ε DMI = Dm · (∂ x m ×x − ∂ y m ×ŷ) [81]. Meanwhile, the sign of DMI constant D designates the chirality of skyrmion-like structures, i.e., the left-handedness or right-handedness. In this work, we focus on the Néel skyrmion, skyrmionium and antiskyrmion, which are realized by considering different DMIs, either the crystallographic class or the strength.
Magnetic parameters. The primary magnetic parameters of Pt/Co magnetic film are adopted [47]: saturation magnetization M s = 580 kA·m −1 , exchange stiffness A = 15 pJ·m −1 , PMA constant K = 0.8 MJ·m −3 in the clean magnetic area without pinning sites. The DMI constant is set to be 3.5 mJ·m −2 , at which both the skyrmion and the skyrmionium are the metastable states corresponding to the minimal values in the energy profile [82]. It should be noted that the oscillation discussed in our work is not limited to this set of parameters, but suitable for a wide range of magnetic parameters. It can be extended to most of the existing materials that support skyrmions.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability
The micromagnetic simulation software OOMMF used in this work is open-source and can be accessed freely at http://math.nist.gov/oommf.

Supplementary information
See the supplementary information for details on the generation of an elongated skyrmion, the conversion between an ordinary skyrmion and an elongated skyrmion, effects of the magnetic parameters on the elongated skyrmion, the stability diagram of elongated skyrmion for different magnetic parameters, effects of the field-like torque on the oscillation frequency, the schematics of the proposed device as a magnonic waveguide and the proposal of a logic gate based on the propagation of spin waves.